Hayat Hanaa Leyla

Khalida

Keywords: Statistiques spatiales, Autocorrélation spatiale, I de Moran, Matrice de poids, Démographie, Mortalité, Analyse des données fonctionnelles, Analyse fonctionnelle en composantes principales, Statistiques non paramétriques, Récidive du suicide, Modèle probit Spatial statistics, Spatial autocorrelation, Moran's I, Weight matrix, Demography, Mortality, Functional data Analysis, Functional principal component analysis (FPCA), Non-parametric statistics, Suicide relapses, Probit model functional principal component analysis, Moran's I, spatial autocorrelation, KNN, contiguity, mortality. Chapter 3 Partially linear spatial probit models Binary choice model, GMM, non-parametric statistics, spatial econometrics, spatial statistics

lorsque l'espace est concerné. En outre, l'utilisation de modèles de régression spatiale pour étudier les récidives de tentatives de suicide, met en évidence l'impact des localisations voisines sur l'acte suicidaire.

Résumé

Ces dernières années, les études sur les prévisions démographiques se sont considérablement développées. L'un des objectifs de la démographie est d'analyser et de prévoir statistiquement les taux de mortalité et de fécondité sans se fier aux opinions subjectives des experts. Par conséquent, pour identifier les caractéristiques de la dynamique de mortalité d'une population, de nombreux modèles ont été développés depuis l'introduction du célèbre modèle proposé par [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF]. De nombreuses recherches disponibles dans la littérature tendent à se concentrer sur la perspective des séries temporelles de la prévision des taux de mortalité. Le manque d'études dans le cadre spatial a suscité notre intérêt pour l'étude des taux de mortalité dans le cadre spatial. L'extension du modèle de Lee-Carter en incorporant l'idée de l'analyse des données fonctionnelles (FDA) a inspiré la première partie de cette thèse où le concept de FDA a été appliqué au cadre d'analyse démographique spatiale. Nous étudions l'existence d'une autocorrélation spatiale pour des données de mortalité des pays voisins. Une méthode fonctionnelle spatiale de composantes principales est proposée pour révéler les modèles spatiaux en considérant directement l'information spatiale. Une statistique fonctionnelle du I de Moran est introduite. Cette statistique aide à déterminer l'autocorrélation spatiale dans les données fonctionnelles par la mise en oeuvre de l'ACP spatio-fonctionnelle. Cette statistique fonctionnelle du I de Moran est la première de son genre dans le cadre des données fonctionnelles.

La deuxième partie de cette thèse étudie l'impact du système VigilanS (programme de prévention des tentatives de suicide en France) sur la récidive suicidaire où les données de ce système (âge, sexe, adresse, historique des tentatives de suicide, séjour à l'hôpital etc.) sont cartographiées sur la carte de la région Nord-Pas-de-Calais tout en construisant des modèles de prédiction spatiale. Les risques de tentatives de suicide sont étudiés à l'aide de modèles probit spatiaux. Nous proposons un modèle probit partiellement linéaire pour les données spatialement dépendantes. Ce modèle n'a pas été étudié dans la littérature d'un point de vue théorique et cette partie comble cette lacune en abordant un modèle d'erreur autorégressive spatiale (SAE) où la structure de dépendance spatiale est intégrée dans un terme de perturbation du modèle étudié. Une méthode d'estimation semi-paramétrique est obtenue en combinant l'approche de la méthode des moments généralisée et la méthode de vraisemblance pondérée. Nous avons examiné l'utilisation de ce modèle de régression probit spatial ainsi que d'autres modèles existants dans la littérature pour étudier les récidives de tentatives suicidaires des patients impliqués dans le système VigilanS.

Cette thèse met en évidence l'importance des statistiques spatiales dans l'analyse des problèmes démographiques et de suicide. Il est également intéressant de voir comment les données fonctionnelles peuvent être utilisées comme un outil dans le domaine de la démographie, notamment pour capturer l'autocorrélation spatiale dans les taux de mortalité

Abstract

In recent years, studies in demographic forecasting have grown significantly. One of the goals of demography is to statistically analyse and predict mortality and fertility rates without relying on subjective opinions of experts. Therefore, to identify the characteristics of the mortality dynamics of a population, many models were developed since the introduction of the famous model proposed by [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF]. Many research available in the literature tend to focus on the time series perspective of forecasting mortality rates. Lack of studies from the spatial framework sparked our interest in investigating the mortality rates from the spatial framework. The extension of the [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF] model by incorporating the idea of functional data analysis (FDA) inspired the first part of this thesis where the FDA concept was applied to the spatial demographic analysis framework. We investigate the existence of spatial autocorrelation in mortality data of neighbouring countries. A functional spatial principal component method is proposed to reveal spatial patterns by directly considering spatial information. A functional Moran's I statistic is introduced. This statistic aids in determining the spatial autocorrelation in functional data through the implementation of the spatio-functional PCA. This functional Moran's I statistic is the first of its kind in the functional data framework.

The second part of this thesis investigates the impact of the VigilanS system (program to prevent suicide reattempts in France) on suicide recidivism where the data from this system (patient's age, sex, address, history of suicide attempts, hospital stay etc.) are mapped on the map of the Nord-Pas-de-Calais region while constructing spatial prediction models. The risks of suicide attempts are mapped with the help of spatial probit models. We propose a partially linear probit model for spatially dependent data. This model has not been investigated in the literature from a theoretical point of view and this part fills that gap by addressing a spatial autoregressive error (SAE) model where the spatial dependence structure is integrated in a disturbance term of the studied model. A semi-parametric estimation method is obtained by combining the generalized method of moments approach and the weighted likelihood method. We examined the use of this spatial probit regression model as well as other existing models in the literature to study the suicide relapses of patients involved in the VigilanS system.

Dedication

'I would like to dedicate this work to GOD Almighty for giving me this opportunity to do this thesis which would not have been possible otherwise.

This work is also dedicated to my father who always prays for me to achieve this work, to the soul of my mother "Hadia", to my brother, my aunt and my husband.'

The first and biggest thank you goes to no one else than my thesis supevisor Sophie Dabo-Niang, who has gone far and beyond the call of duty from time to time, to push me forward, keep me in the game, and simply was there all the time despite her various responsibilities and her busy schedule. We share this journey together from the master's study till the third year. Thank you for having given me your confidence by accepting to supervise my thesis but also for having given me the chance to live enriching experiences. I hope that we will have the opportunity to collaborate during the next years.

I am very grateful to Dr Dharini Pathmanathan for her constant guidance and supervision, as well as for providing necessary information regarding the project. Her constant guidance and willingness to share her vast knowledge helped me to complete the assigned tasks on time. I would like to thank you for pushing me forward and for being there all the time despite your various responsibilities, your busy schedule, and the time lag between France and Malaysia.

I would like to thank the members of the jury who agreed to judge this work and to be the rapporteurs of my thesis and who agreed to examine my project and to be part of this jury. My thanks and appreciations also go to Marielle and Zeinab. I thank the doctoral students of the laboratory Ahmed Salem, Zied, Fahriat, Salsabil, Solange, Yoba, Mamadou, Feryel and Christelle. My friends with whom I spent beautiful moments and unforgettable memories, thank you for being my family in France. I love you all very much: Fatima, Zaynab, Doha, Hussein,Jana, Kawthar, Meryem, Batoul, Zeinab, Joelle, Abir and all the girls. Thank you to my cousin Sarah and her husband Maysam who have always been here for me and to hear my news. I also thank Ghadir and Zakariya and their little girls Sara and Tasnim for being my family in Lille. I would like to thank all my family: téta, my cousins, my uncles and aunts especially

Introduction

This thesis is motivated by two real population health problems such as mortality forecasting and the mapping of suicidal relapses.

The first part of this thesis investigates mortality from a spatial perspective. In recent years, studies of demographic forecasts have grown significantly. One of the goals in demographic research is to analyse and predict mortality and fertility rates in a purely statistical way without relying on the subjective opinions of experts. Therefore, to identify the characteristics of the mortality dynamics of a population, many models have been developed since the introduction of the famous model proposed by [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF]. Many researches available in the literature tend to focus on time series forecasting of mortality rates. However, this research sparked the interest in investigating mortality from the spatial framework since the main theme of this thesis is spatial statistics. The extension of the [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF] model by [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] using functional data analysis (FDA) inspired the application of the FDA concept in the spatial framework.

In this thesis, we investigate the existence of spatial autocorrelation in mortality data of neighbouring countries. The spatial principal component analysis by [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] aimed to reveal spatial patterns by directly considering spatial information. We constructed a functional Moran's I statistic based on this, which will aid in determining spatial autocorrelation in functional data through the implementation of the spatio-functional PCA. This functional Moran's I statistic is the first of its kind in the functional data framework. The data of mortality rates for 28 European countries from the Human Mortality Database (HMD) for ages 0 to 110 (where ages above 100 are grouped as 100 +) were used to investigate the spatial dependency at a fixed time, where years 1990, 2000 and 2010 were examined separately).

The second part investigates the area of spatial modelling by considering a partially linear probit model for spatially dependent data. Semi-parametric binary models have not been investigated from a theoretical point of view and this part fills that gap by addressing a spatial autoregressive error (SAE) model when the spatial dependence structure is integrated in a disturbance term of the studied model. A semi-parametric estimation method is obtained by combining the generalized method of moments approach and the weighted likelihood method.

In France, the VigilanS healthcare system is an effort to support those who have attempted suicide in various regions. It was established in Nord-Pas-de-Calais in February 2015. This programme to monitor and prevent recidivism of suicide attempts is executed via phone calls by teams of professionals who are specialized in this type of remote care. This six-month programme is managed by the adult psychiatry department, directed by Professor Guillaume Vaiva in the Lille University Hospital. Posthospital support is offered to those patients who attempted suicide.

xii Chapter 0. Introduction

The patients receive a resource card with a dedicated contact number and are called back for follow up after 6 months. Those with a history of suicide reattempts are contacted from the 10th to 21st day, after the suicide attempts. The VigilanS healthcare database of the Lille University Hospital contains the data of more than 23000 patients which include age, sex, address, history of recidivism of suicide attempts, hospital stay (date, duration of hospitalization, establishment), recidivism of suicide attempt at the first enrolment (mode, contact, alcohol consumption). The aim of this study is to investigate recidivism of suicide attempts for all patients at the initial point of enrollment and after 6 months. It is of interest to study the spatial dependency of the occurrences of recidivism of suicide attempts. Furthermore, the risks of suicide attempts are mapped with the help of the spatial probit models. The impact of the VigilanS system on recidivism of suicide attempts is investigated, where the data from this system (patient's age, sex, address, history of suicide attempts, hospital stay etc.) are mapped on the map of Nord Pas de Calais while constructing spatial prediction models. The factors of suicide relapse among these patients based on the regions they belong to were studied.

Chapter 1 discusses the fundamental concepts and a state of art of the models and methods used. In Chapter 2 of this thesis, we examined the idea of FPCA which is an important dimensionality tool reduction technique on functional data with infinite dimensionality. We investigated the spatial autocorrelation of mortality rates for 28 European countries, with data from the Human Mortality Database (HMD) using spatial associations in the context of functional areal data. This motivated to the development of a functional Moran's I statistic which is the first of its kind in the functional data analysis framework, which can detect spatial autocorrelation and spatial PCA for areal data.

Chapter 3 focuses on partially linear spatial probit models. Here we are interested in various patterns of spatial data by assuming conditional spatial heteroscedasticity, non-identically distributed observations, and a linear process for disturbances. The estimation procedure involves the combination of a weighted likelihood and a generalised method of moments.

Chapter 4 is mainly an application of the models studied in Chapter 3. Here, we investigate the impact of the Vigilans system (programme to prevent suicide reattempts in France) on suicide recidivism where the data from this system (patient's age, sex, address, history of suicide attempts, hospital stay etc.) are mapped on the map of the Nord-Pas-de-Calais region. This will aid in making spatial prediction models. The probit regression model was used to study the suicide relapses and then the risk of suicide attempts was mapped by using the spatial probit and spatial Poisson models.

Chapter 5 provides the concluding remarks as well as some significant contributions of this research. Suggestions on extending research works related to this research are also included in this chapter.

Chapter 1

State of art and general concepts

This chapter discusses the methods used in formulating the main outcomes of this research.

Spatial data analysis and spatial modelling are the main components of this thesis. We implement the use of functional data analysis in the field of demography with respect to mortality models to investigate spatial dependence of mortality rates of neighbouring countries. The second part of this study focuses on examining the theoretical foundations and application of spatial econometric models.

Functional data analysis

Functional data analysis (FDA) has been gaining importance in analysis of data involving surfaces, multidimensional objects, shapes or more complex mathematical objects of infinite dimension [START_REF] Cardot | Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data[END_REF] . [START_REF] Ramsay | Functional data analysis[END_REF] gave an excellent overview of the concepts, foundation and applications of FDA. Some reputable works in this field include those of [START_REF] Bosq | Linear processes in function spaces[END_REF], [START_REF] Ferraty | Nonparametric functional data analysis [Theory and practice[END_REF], [START_REF] Horváth | Inference for functional data with applications[END_REF] and [START_REF] Hsing | Theoretical foundations of functional data analysis, with an introduction to linear operators[END_REF].

Functional data is applied widely in various fields such as medicine, meteorology, hydrology, genetics (genetic sequence) and so on. FDA deals with the analysis and theory of data that are in the form of functions, rather than by vectors of R n . In other words, each observed variable has functional values rather than real values.

A functional random variable is a random variable which takes its values in a vector space of infinite dimension, and the functional data is a realization of a functional variable. The functional data are considered as observations of infinite-dimensional stochastic processes. Let I be a compact interval of R d with d ∈ N, we observe N independant realizations X (1) , . . . , X (n) , . . . , X (N ) of an underlying stochastic process X = (X t : t ∈ I), where X (n) is a continuous function on I. Random functions can be viewed as random elements taking values in a Hilbert space L 2 (I).

The associated inner product is defined as:

f, g = I f (t)g(t)dt, f, g ∈ L 2 (I). (1.1)
The mean function µ : I → R and the covariance function c : I × I → R are defined as:

µ(t) = E(X(t)), c(s, t) = Cov[X(s), X(t)], s, t ∈ I. (1.2)
Chapter 1. State of art and general concepts

The statistical methods for multivariate data encounter difficulties involving high dimension.

This is mainly because handling infinite dimensions of functional variables (curves, shapes, and so on) can be complicated. Besides that, the dependency between observations when considering time-series of spatial functional objects may be difficult to manage. Therefore, it is necessary to develop statistical methods for visualization and modeling to handle such data. During the last two decades, various exploration and modeling techniques have been proposed for functional variables. As for regression models, there are essentially two popular approaches: the parametric [START_REF] Ramsay | Functional data analysis[END_REF] and nonparametric models [START_REF] Ferraty | Nonparametric functional data analysis [Theory and practice[END_REF]. [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] proposed the combination of ideas from FDA, nonparametric smoothing and robust statistics to form a methodology which is a significant contribution to the field of demography especially mortality modeling. This idea is widely applicable to any functional time series data, particulary age-specific mortality and fertility. A nonparametric smoothing method was used to smooth the data and the fitted curves were decomposed via a basis function expansion. This method is discussed further in Section 1.2.2.

From discrete data to functions

The first step in FDA is to convert a discrete set of measurements (the observed data points) into either a rough or smooth curve.

Let Φ = {φ j (•) : j ∈ N} be an infinite basis of L 2 (I). The elements of Φ are linearly independent.

Every element of L 2 (I) can be written as a linear combination of the elements of Φ. The realization X (n) of the stochastic process X is decomposed into:

X (n) (•) = j≥1 c j,n φ j (•), (1.3)
where {c j,n } j≥1 is an infinite set of coefficients. The basis expansion is used to approximate the realization X (n) by its projection on the span of a finite basis functions

Φ J = {φ j (•) : 1 ≤ j ≤ J},
a finite subset of Φ and {c j,n } 1≤j≤J a subset of {c j,n } j≥1 : n) can be summarized by a J-dimensional vector. The functional data smoothing methods imply a decomposition of each of the process realizations into a common basis of function, such as the Fourier bases or splines bases. The Fourier basis system is commonly used for periodic data while the B-spline basis system is preferable for nonperiodic data [START_REF] Ramsay | Functional data analysis with r and matlab: Springer science & business media[END_REF].

X (n) (•) ≈ J j=1 c j,n φ j (•), (1.4) X (
Splines are polynomial segments joined end-to-end and forced to be smoothed at the joint.

Let {t i , X i : i = 1 . . . n} be a set of observations X i at design points t i , modeled by the relation

X i = s(t i ) + i
where s(t) is a smooth curve and i are iid. The s(t) that minimizes the residual sum of squares (RSS) plus roughness penalty is the cubic smoothing spline fit to the data:

n i=1 (x(t i ) -s(t i )) 2 + λ s (t) 2 dt, (1.5)
1.1. Functional data analysis

3
where λ ≥ 0 is the smoothing parameter which governs the trade-off between smoothness and goodness of fit.

Functional principal component analysis

Functional principal component analysis (FPCA) is the extension of the multivariate principal component analysis in the functional framework. As in the classical case, the FPCA corresponds to an optimal linear representation of a set of functional data in a finite dimensional space. This is to reduce the dimensionality of the data using FPCA and then identify the main sources of variability. The point behind FPCA which is a dimension reduction method, is transforming the sampled curves so that only a low-dimensional space represents the patterns of variability of the curves. The general approaches are presented by [START_REF] Ramsay | Functional data analysis[END_REF] and [START_REF] Ferraty | Nonparametric functional data analysis [Theory and practice[END_REF].

If we have n functional observations in L 2 , X (1) , . . . , X (n) , we thus look for j functions of L 2 , φ 1 , . . . , φ k , orthogonal, and such that the projection of X (i) onto the vector space generated by the φ j generates the minimum loss possible. The FPCA is performed by searching the spectrum of a compact operator. This operator is defined from the covariance function given by :

c(s, t) = Cov[X(s), X(t)], = 1 n n i=1 (X i (s) -µ(s))(X i (t) -µ(t)), (1.6)
where µ denotes the average function of X i . FPCA identifies principal components explaining the variability of {X i } by computing the eigenfunctions corresponding to the ordered eigenvalues (from largest to smallest) of an empirical covariance operator. Therefore, performing the PCA of the X i amounts to searching for the eigenvalues of the operator Γf (t) = L 2 (I) -→ L 2 (I) defined by:

Γf (t) = C(•, t), f (•) , t ∈ I, f ∈ L 2 (I), (1.7)
where Γ is a positive, linear and self-adjoint operator in L 2 [START_REF] Horváth | Inference for functional data with applications[END_REF]. In particular, it is a compact operator and has a finite trace.

There exists a complete orthonormal basis {φ j } j≥1 and a sequence of real numbers

λ 1 ≥ λ 2 ≥ • • • ≥ 0 such that: Γφ j = λ j φ j , and λ j → 0 as j → ∞, (1.8)
where {λ j } j≥1 is the set of eigenvalues of the covariance operator Γ associated to {φ j } j≥1 the set of its eigenfunctions. The eigenfunctions associated to the eigenvalues are then the {φ j }. It can be shown that the eigenfunction associated with the largest eigenvalue, φ 1 , is a solution of the following constrained optimization problem:

max φ 2 =1
Γφ, φ (1.9)

The process X admits the Karhunen-Loève representation:

X(t) = µ(t) + j≥1 c j φ j (t), t ∈ I, (1.10) with c j = X -µ, φ j , E(c j ) = 0, cov(c j , c l ) = λ j 1
j=l and the {φ j } j≥1 are the FPCA basis.

Chapter 1. State of art and general concepts

Hence, X (n) is approximated by truncating the infinite sum at the first J terms:

X (n) (t) ≈ µ(t) + J j=1 c j,n φ j (t), t ∈ I with c j,n = X (n) -µ, φ j .
(1.11)

The concept of FPCA in FDA is the crux of the analysis of mortality data. The classical FPCA was performed to find new functions that reveal the most important variations in the curve data with the absence of the spatial structure. After that, this idea was extended to the spatial framework with the implementation of the FPCA on areal spatial data.

1. [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF]) is one of the most prominent contributions to the study of mortality models which opened the path to various innovations of its kind. The Lee-Carter model is given as follows:

log(m xt ) = a x + b x k t + x,t
(1.12) where • m x,t is the age-specific death rate for age x and year t;

• a x is the average age of the log mortality rates across years;

• b x is a deviation in mortality due to changes in the time index (k t );

• k t is the mortality index in the year t; it describes the evolution of the level of mortality over time.

• the error term x,t reflects age-specific historical influences not captured by the model.It is normally distributed with mean 0 and variance σ 2 .

Three unboserved parameters a x , b x and k t in the single equation 1.12 means that the LC model is over-parametrized and therefore two normalization constraints are imposed:

k t = 0, b x = 1.
The parameters b x and k t are obtained by singular value decomposition (SVD). [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF] fitted the parameter k t using standard ARIMA models. The random walk model with drift given as follows:

k t = k t-1 + d + e t ,
was found to be suitable to describe k t , where d is the drift parameter which reflects the average annual change and e t is an uncorrelated error.

Some of the commonly used extensions of the LC model consist of the [START_REF] Renshaw | A cohort-based extension to the lee-carter model for mortality reduction factors[END_REF] model which used the generalised linear model approach, the [START_REF] Lee | Evaluating the performance of the lee-carter method for forecasting mortality[END_REF] which involves re-estimation of the time component according to the observed life expectancy at birth, and the [START_REF] Booth | Applying lee-carter under conditions of variable mortality decline[END_REF] The HU model is more robust than the LC model when we consider the variation explanation, outlier identification and forecast accuracy. The HU method extends the LC method by :

• smoothing the log mortality rates before modeling,

• using the FPCA,

• using more than one principal component for forecasting. [START_REF] Hyndman | Stochastic population forecasts using functional data models for mortality, fertility and migration[END_REF] used functional data models with time series coefficients to model age-specific mortality and fertility rates. [START_REF] Hyndman | Coherent mortality forecasting: The product-ratio method with functional time series models[END_REF] proposed a method for nondivergent or coherent forecasting of mortality rates for two or more subpopulations, based on functional principal components models of simple and interpretable functions of rates. In a nutshell, various approaches in forecasting mortality rates were discovered using functional data analysis as a foundation.

The HU method is summarized in three steps:

1. Before modeling, [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] proposed to smooth the log death rates using penalized regression splines. Assume that there is a smooth continuous underlying function S t (x) which is observed with error at discrete ages.

Y t (x i ) = S t (x i ) + σ t (x i ) t,i ; t = 1 . . . n; i = 1 . . . p (1.13)
where Y t (x i ) denotes the log of observed death rates for age x i at year t, S t (x i ) is the derived smooth function of x, σ t (x i ) is a noise component enables the amount of noise to vary with x in year t, hence correcting the assumption of homoscedastic error in the LC model, and the t,i is an independent and identically distributed standard normal random variable.

Estimation of the smoothing function S t (x) is done through x, while y t (x) by a nonparametric smoothing method ∀t.

2. The FPCA on the time series was applied to find the main sources of variability. A larger number of principal components were used to capture additional dimensions of changes in mortality rates. The FPCA is expressed as :

S t (x) = µ(x) + K k=1 β t,k φ k (x) + e t (x), (1.14)
where S t (x) is the derived smooth function of x, µ(x) is the mean function estimated by 1.2. Mortality modeling 7 3. We are interested to forecast the future functional data h-years ahead. Therefore, the main idea is to use the obtained principal component scores to predict new values of βn,k,h , hyears ahead conditioned to the fixed set of functional principal components φ k (x). The estimated coefficients are then used to approximate the functional data using equation (1.14). Hence, the h-step ahead forecast of Y n+h (x) is obtained by:

μ(x) = 1 n n 1 S t (x), {φ k (x)} = {φ 1 (x), . . . , φ k (x)} is the set of first K functional principal components (orthonormal basis function), {β k } = {β 1 , . . . , β k } is
Ŷn,h (x) = E[Y n+h (x)] = μ(x) + K k=1 βn,k,h φ k (x);
(1.15)

where βn,k,h is the h-step ahead forcast of β n,k,h . The coefficicients, βt,k and βt,l are uncorrelated for k = l so the univariate time series model such as ARIMA is used to predict the time-varying scores {β t,k }.

The HU method can explain more variation of the demographic dynamics when we have data of high quality. However, its performance is comparable to the LC model when limited and scarce data sets such as Chinese data sets are used [START_REF] Fang | Stochastic population analysis: A functional data approach[END_REF].

Spatial data analysis

Statistics for spatial data was first explored in the 1960s in geology and meteorology. The main feature of spatial data analysis is to find a correlation structure between data observed at a given location and that available at neighbouring locations. These kind of data are widely available in economics, epidemiology, agriculture, environmental science and so on.

The past five decades witnessed the development of several methods in analysing spatial data and estimating values of a property of interest for the unsampled locations, from the available sample data. Data dependency is one of the practical considerations that influence the available techniques used in the spatial data modeling. Often, spatial data are dependent and a spatial model is tailored to handle this aspect. The realisation of the importance of spatial data analysis led to the development of statistical models to capture the spatial patterns [START_REF] Cressie | Statistics for spatial data[END_REF]. A wide range of models and methods have been developed by considering the spatial dependence structure, mainly in the context of geostatistics, lattice data, and point patterns [START_REF] Cressie | Statistics for spatial data[END_REF].

In the framework of geostatistics, the spatial location is valued in a continuous set of R N , N ≥ 2.

When compared to geostatistical and lattice data, spatial point patterns occurs when the locations of the available data are random. It is not always easy to distinguish the following three types of data:

1. Geostatistical data

• The spatial set of interest S ⊂ R N , N ≥ 2 is a fixed subset of the plane of positive area (2-D) or volume (3-D).

• A spatial process (collection of random variables observed at spatial points) Y = {Y (s), s ∈ S} is of interest.

Lattice data

• The spatial process Y = {Y (s), s ∈ S} of interest is defined on a spatial fixed regular or irregular lattice S of R N .

• This type of process includes extensions to well know time-series process.

Point patterns

• The spatial locations s ∈ S ∈⊂ R N where the process Y = {Y (s), s ∈ S} is defined are random.

• This type of process is an extension of the usual point processes.

The major distinction between time series and spatial data is the absence of an orderly relationship such as past, present and future i.e. the time axis is unidirectional. Past events may have an influence on the future while the reverse is not true. Time series models are not applicable to spatial data directly since the natural order in the time domain does not exist in the spatial context.

In this thesis, we are interested in the spatial lattice processes. For spatial lattice data, the locations form a lattice set. For lattice data, the seminal work by [START_REF] Besag | Spatial interaction and the statistical analysis of lattice systems[END_REF] is of great significance. The underlying idea is that a lattice has a neighbourhood structure and observations recorded on a lattice point are conditionally independent of the remaining lattice points given 1.3. Spatial data analysis the observation in the neighbouring lattice points [START_REF] Kauermann | A hitchhiker's view on spatial statistics and spatial econometrics for lattice data[END_REF]. A comprehensive collection of applications and theory in this field are available in [START_REF] Rue | Gaussian markov random fields: Theory and applications[END_REF] as well as [START_REF] Gaetan | Spatial statistics and modeling[END_REF]. Similar to other spatial processes, the exploration of the spatial correlation structure is the first step in the context of the lattice. The spatial weight matrix is a basic correlation tool in spatial econometrics describing the connectivity between different locations. The spatial weight matrices take different forms.

Specification of the spatial weight matrix

Spatial autocorrelation and its modeling are essentially based on the spatial weight matrices.

In the spatial econometrics's literature, spatial dependency between spatial units is defined via the spatial weight matrix. W n is a n × n non-stochastic weight matrix, it describes the spatial interactions between the n spatial units.

Formally, W n is a positive n × n matrix with zero on the diagonal:

W n =              0 w 1,2 • • • w 1,j • • • w 1 , n w 2,1 0 • • • w 2,j • • • w 2 , n . . . . . . . . . • • • • • • • • • w i,1 w i,2 . . . 0 • • • w i,n . . . . . . . . . . . . . . . • • • w n,1 w n,2 . . . . . . . . . 0             
where w i,j is the spatial weight between locations i and j.

The elements w ij = w ij,n of this matrix are usually considered as inversely proportional to the distance between spatial units i and j with respect to some metric (physical distance, social network or economic distance, [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. More specifically, the weight matrices are classified into two groups: weights based on distance and weights based on boundaries. For weights based on distance, spatial weight matrices are constructed by using the distance d ij between each pair of spatial units (regions, cities, centroids, . . . ) i and j. Weights based on distance are as follows:

• k-Nearest Neighbour weights

w ij = 1 if j ∈ N k (i),
0 Otherwise, where N k (i) is the set of the k closest units or regions to i for k ∈ {1, ..., n -1}

• Radial Distance weights

w ij = 1 if 0 ≤ d ij ≤ δ, 0 if d ij > δ
, where d ij is the Euclidean distance between units i and j, and δ is a critical distance (threshold distance or bandwidth) cut-off after which spatial effects are considered to be negligible, and it should be able to guarantee that each region has at least one neighbour.

• Power Distance Decay weights

w ij = d -α ij if 0 ≤ d ij ≤ δ, 0 if d ij > δ, where α is any positive exponent, typically α = 1 or α = 2.
Chapter 1. State of art and general concepts

• Exponential Distance Decay weights

w ij = exp(-αd ij ) if 0 ≤ d ij ≤ δ, 0 if d ij > δ,
• Double-Power Distance weights

w ij = [1 -(d ij /δ)] k if 0 ≤ d ij ≤ δ, 0 if d ij > δ, with k is a positive integer, typically k = 2, k = 3 or k = 4.
• Cliff-Ord weights [START_REF] Cliff | Spatial autocorrelation[END_REF] suggested to use the length of the common border between contiguous regions, weighted by a distance function:

w ij = d -a ij D b ij ,
where D ij is the share of common boundary between i and j. a and b are parameters estimated from data or chosen a priori.

• Block structure

In this case w ij = 1 for all i and j in the same block. The blocks are defined according to some specific criterion.

For weight matrices based on boundaries, the spatial contiguity is often used to specify neighbouring locations that share a common boundary. Various spatial contiguities are available in the literature. In the classical case of a regular square grid layout, the options of contiguity are referred to as the Rook contiguity (only common boundaries), Bishop contiguity (with only common vertices) and Queen contiguity (both boundaries and vertices). The contiguity weights are given as follows:

w ij = 1 if i and j are contiguous, 0 otherwise.
In general, the last equation can be rewritten as:

w ij = 1 ij > 0,
0 ij = 0, where ij denotes the length of shared boundary.

The Moran's index

The basic principle of spatial data analysis is the idea that values of variables in nearby locations are closely related compared to those locations that are far apart. The Moran's index is a measure of spatial autocorrelation. It was introduced first by Moran (1948,1950) and developped by Cliff andOrd (1973, 1981). Moran's I is a correlation coefficient used to measure the overall spatial correlation in a data set and is bounded by 1 and -1. The spatial autocorrelation in terms of Moran's I can be classified as follows:

• positive autocorrelation occurs when Moran's I is close to +1.

The spatial correlation is positive when similar values cluster together on a map.

• negative autocorrelation occurs when Moran's I is close to -1.

The spatial correlation is negative when dissimilar values cluster together on a map.

• a Moran's I value of 0 denotes the absence spatial autocorrelation.

The results of this test is interpreted in the context of a null hypothesis which assumes that the random of distributed among locations. Suppose that, in at n locations {s 1 , . . . , s n }, we observe y 1 = y(s 1 ); . . . ; y n = y(s n ). We also suppose that spatial weights w ij , the weights between each pair of spatial units s i and s j , which satisfy:

w ij ≥ 0 for any i = j,
w ii = 0 for any i.

The Moran's I index is given by:

I = n W n i=1 n j=1 w ij (y i -ȳ)(y j -ȳ) n i=1 (y i -ȳ) 2 , (1.16)
where n is the number of spatial units indexed by i and j, ȳ is the mean of y and w ij is a matrix of spatial weights with zeroes on the diagonal and W is the sum of all w ij .

In this thesis, the Moran's I is extended to the functional data analysis context which was discussed in Section 1.1. This aims to consider the spatial dependency in the PCA to analyze the degree of spatial autocorrelation among observations in the geographic space with the idea of spatial principal component analysis, sPCA [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] which highlights spatial patterns by the direct consideration of spatial information.

Functional PCA for spatial data

There are many functional data sets taking the form : X s (t), s ∈ S and t ∈ [0, T ]. where s is a spatial location and [0, T ] is the interval of time. In such data, the spatial dependence are often not taken into account. Note the famous example of the Canadian temperature data [START_REF] Ramsay | Functional data analysis[END_REF] [START_REF] Li | Functional principal component analysis of spatiotemporal point processes with applications in disease surveillance[END_REF]; [START_REF] Liu | Functional principal components analysis of spatially correlated data[END_REF]; [START_REF] Hörmann | Consistency of the mean and the principal components of spatially distributed functional data[END_REF]; [START_REF] Liu | Functional principal components analysis of spatially correlated data[END_REF]; [START_REF] Illian | Principal component analysis for spatial point processes -assessing the appropriateness of the approach in an ecological context[END_REF]). [START_REF] Gromenko | Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends[END_REF][START_REF] Gromenko | Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves[END_REF] use a geostatistical framework for spatially indexed functional data to solve space physics problems. [START_REF] Kuenzer | Principal component analysis of spatially indexed functions[END_REF] proposed a dimension reduction technique suitable for functional data, indexed by spatial locations on a grid. So far, there is no work on FPCA on spatial data in the the lattice framework.

The idea of the HU model in equation (1.13) motivated the use of functional data analysis in our study. The versatility of functional data analysis is appealing because it can be adapted to many areas of mortality forecasting. The smoothed data are ready for conversion to functional objects and after that various approaches can be implemented so these data can be investigated from different angles. This thesis investigates mortality from a spatial point of view instead of the usual approaches which give emphasis to the time aspect of mortality models. It is of interest to investigate the existence of spatial autocorrelation among neighbouring countries by adapting dimensionality reduction techniques using FPCA.

The sPCA by [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] aimed to reveal spatial patterns by directly considering spatial information. The sPCA analyses a matrix of relative allele frequencies X where the spatial information is stored inside a spatial weighting matrix L, and defines the following function to measure both the spatial structure and variability in x:

C(x) = var(x) I(x) = 1 n x T Lx.
(1.17)

Interestingly, spatial autocorrelation can efficiently be captured from functional data using the newly derived functional Moran's I statistic in this thesis. This statistic is the first of its kind in the functional data framework. 28 European countries whose mortality data are available on HMD, for a fixed period since it is of interest to investigate the existence of spatial autocorrelation in mortality data of neighbouring countries are considered. Once this approach successfully captures the spatial autocorrelation, it will be of great help to use this idea to construct spatial predictive models to predict mortality rates for neighbouring countries whose data are unavailable on HMD.

Spatial econometric models

Modelling spatially dependent data requires correlation between random variables in one location with those in neighbouring locations [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. In this thesis, the lattice type data is examined. Statistical models for lattice data are linked to nearest neighbours to express the fact that data are nearby. Two popular spatial dependence models for lattice data are the spatial autoregressive (SAR) dependent variable model and the spatial autoregressive error model (SAE, where the model error is an SAR), which extend the regression in a time series to spatial data.

Let (Y, X) be a random vector observed at n locations where {s 1 , . . . , s n } in an irregularly spaced, countable lattice

I ⊂ R k , k ≥ 2 such that ||s i -s j || ≥ d 0 , with d 0 > 0. Suppose that Y n = (Y 1 , . . . , Y n ) T
is the sample response and X n the n × p matrix of explanatory variables observations with elements X ij , i = 1, . . . , n, j = 1, . . . , p.

To explain why an observation located in a specific location depends on observations made at other locations, it is necessary to determine the three types of interaction effects:

• Endogenous interaction effects among dependent variables. The variable Y i at spatial units i depends on Y j at spatial units j. This model is named as the spatial lag model or spatial autoregressive (SAR) model [START_REF] Cliff | Spatial autocorrelation[END_REF],where the interaction effect is denoted by the spatial lag W n Y n .

• Exogenous interaction effects among independent variables where the variable Y i at a spatial unit i depends on the independent explanatory variables X j at spatial units j.

• Correlated effects, where similar unobserved characteristics result in similar behavior.

Here, the interaction is among the error terms. This model is known as the spatial autoregressive error (SAE) model (or spatial error model; SEM).

In practice, a population that contains these three types of interactions jointly does not exist.

Researchers always focuses on models with one interaction such as the SAR model, the SAE model, or a model with two interactions. These kind of models are used when the spatial autocorrelation affects the response and the error terms. According to the terminology developed by [START_REF] Lesage | An introduction to spatial econometrics[END_REF], we refer to this model as the spatial autocorrelation (SAC) model:

1.3. Spatial data analysis 13

Y n = λ 0 W n Y n + X n β 0 + U n ; U n = γ 0 W n U n + ε n , ε n ∼ N (0, σ 2 0 I n ),
(1.18)

where U n = (U 1 , . . . , U n ) T and ε n = (ε 1 , . . . , ε n ) T . The coefficients λ 0 and γ 0 are scalar autoregressive parameters indicating the degree of spatial dependence, β 0 is a p×1 vector of parameters.

W n Y n is the spatial lag, which denotes the endogenous interaction effects among the dependent variables, i.e. for each observation Y i , the corresponding element in W n Y n gives weighted sum of Y j , j = i, with weights given by the relative connectivity from j to i. W n U n represent the interaction effects among the disturbance terms of the different spatial units. However, the SAR model is a SAC model with γ 0 = 0 and SAE model is a SAC model with λ 0 = 0.

From a theoretical point of view, various linear spatial regression SAR and SAE models as well as their identification and estimation methods, e.g., two-stage least squares (2SLS), three-stage least squares (3SLS), maximum likelihood (ML) or quasi-maximum likelihood (QML) and the generalized method of moments (GMM), have been developed and summarized by many authors such as [START_REF] Anselin | Spatial econometrics: Methods and models[END_REF], [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF], [START_REF] Kelejian | A generalized moments estimator for the autoregressive parameter in a spatial model[END_REF], [START_REF] Conley | Gmm estimation with cross sectional dependence[END_REF], [START_REF] Cressie | Statistics for spatial data[END_REF], [START_REF] Case | Spatial patterns in household demand[END_REF], L.-F. [START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF], L.-f. [START_REF] Lee | GMM and 2SLS estimation of mixed regressive, spatial autoregressive models[END_REF], [START_REF] Lin | GMM estimation of spatial autoregressive models with unknown heteroskedasticity[END_REF], [START_REF] Zheng | On the asymptotics of maximum likelihood estimation for spatial linear models on a lattice[END_REF], [START_REF] Malikov | Semiparametric estimation and testing of smooth coefficient spatial autoregressive models[END_REF], [START_REF] Garthoff | Control charts for multivariate spatial autoregressive models[END_REF], [START_REF] Yang | Identification and QML estimation of multivariate and simultaneous equations spatial autoregressive models[END_REF].

Semi-parametric modeling

A semi-parametric model is an alternative to fully parametric models when there exists a nonlinear relationship between the discrete binary variable and some explanatory variables. This type of model is known as the partially linear probit model for spatially dependent data. A triangular array setting is used to cover various patterns of spatial data. Allowing various spatial dependencies, we assume the existence of conditional spatial heteroscedasticity, non-identically distributed observations, and a linear process for disturbances.

We consider that at n spatial locations {s 1 , s 2 , . . . , s n } satisfying s i -s j > ρ with ρ > 0, observations of a random vector (Y, X, Z) are available. Assume that these observations are considered as triangular arrays [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF] and follow the partially linear model of a latent dependent variable Y * :

Y * in = X T in β 0 + g 0 (Z in ) + U in , 1 ≤ i ≤ n, n = 1, 2, . . . (1.19) with Y in = I (Y * in ≥ 0) , 1 ≤ i ≤ n, n = 1, 2, . . . ( 1
U in = λ 0 n j=1 W ijn U jn + ε in , 1 ≤ i ≤ n, n = 1, 2, . . . (1.21)
where λ 0 is the autoregressive parameter, valued in the compact subset Θ λ ⊂ R, W ijn , j = 1, ..., n are the elements in the i-th row of a non-stochastic n×n spatial weight matrix W n , which contains the information on the spatial relationship between observations. This spatial weight matrix is usually constructed as a function of the distances (with respect to some metric) between locations [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF].

The estimation procedure is a combination of a weighted likelihood [START_REF] Staniswalis | The kernel estimate of a regression function in likelihood-based models[END_REF]) and a generalised method of moments [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. The procedure first fixes the parametric components of the model and then estimates the non-parametric part using weighted likelihood;

the obtained estimate is then used to construct a GMM (Generalised Method of Moments) parametric component estimate. The consistency and asymptotic distribution of the estimators are established under sufficient conditions. Some numerical results are provided to investigate the finite sample performance of the estimators.

In Chapter 3 of this thesis, a semi-parametric estimation method combining the GMM approach and the weighted likelihood method is proposed. The parametric components of the model are first fixed and the non-linear components are estimated by the weighted likelihood approach [START_REF] Staniswalis | The kernel estimate of a regression function in likelihood-based models[END_REF]. The estimator which is obtained depends on the values at which the parametric components are fixed. This is used to construct a GMM estimator [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF] of these components.

In chapter 4, spatial probit regression models are applied to study suicidal relapses to shed light on the impact of neighbouring locations on suicide cases by investigating the impact of the Vig-ilanS system on suicide relapses of patients involved in the VigilanS system.

The VigilanS healthcare system is an effort to support those who have attempted suicide in various regions. It was established in Nord-Pas-de-Calais in February 2015. This program to monitor and prevent recidivism of suicide attempts is executed via phone calls by teams of professionals who are specialized in this type of remote care. In the Lille University Hospital, this six-month programme is managed by the adult psychiatry department under Professor Guillaume Vaiva.

Posthospital support is offered to those patients who attempted suicide. When these patients are discharged from the hospital, they receive a resource card with a dedicated contact number and are called back for follow-up after 6 months. Those with a history of suicide reattempts are contacted between the 10th to 21st day after being discharged from the hospital.

We aim to investigate the impact of spatial dependency on suicide recidivism in Nord-pas-de-Calais and examine the effect of non-linear explanatory variables such as median revenue, unemployment rates etc by applying several frameworks of the probit regression to model the suicide recidivism, especially after 6 months from the entry of VigilanS system.

Introduction

Mortality rates depict the population health and economic status of a country. Spatial demographic models play a major role in monitoring spatial dependence of mortality. Demographers, social scientists, economists, and many others have been extensively studying mortality patterns. Most researches available in the literature are inclined to mortality models with emphasis on temporal forecasting methods. Hence, we were motivated in investigating mortality from the spatial framework.

The Lee-Carter (LC) model [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF]) is one of the most prominent contributions to the study of mortality models which opened the path to various innovations of its kind. The For a fixed country, [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] applied FPCA to decompose smoothed functional time series into a set of functional principal components and their principal component scores. FPCA was applied to find the main sources of variability. [START_REF] Léger | What can we learn from the functional clustering of mortality data? an application to the human mortality database[END_REF] showed that a functional framework can be informative where it allows clustering of complete mortality profiles without losing sight of the role played by single components where the changes of age-specific mortality in low-mortality countries in the last few decades with functional clustering were investigated. [START_REF] Léger | What can we learn from the functional clustering of mortality data? an application to the human mortality database[END_REF] suggested three different methods of functional clustering of mortality profiles (seen as curves over age, which can be observed for every country and every year) by: a two-stage method based on spline coefficients, a distance-based method through FPCA and a model-based method.

In this paper, we use the functional data analysis approach since it works with smooth curves rather than scalar data. We aim to investigate the spatial relationship of mortality of

Description of data

I statistic which will aid in determining spatial autocorrelation in functional data through the implementation on the spatio-functional PCA. Our functional Moran's I statistic is the first of its kind from the functional data framework as no work has been done so far from this perspective.

Section 2.2 gives a brief description of the data used, Section 2.3 explains the methodology used, Section 2.4 discusses the results and Section 2.5 concludes.

Description of data

We consider mortality rates for 28 European countries (Figure 2.1) available on HMD for ages 0 to 110 (where ages above 100 are grouped as 100+) to investigate the spatial dependency at a fixed time, where years 1990, 2000 and 2010 were investigated separately. [START_REF] Lee | Modeling and forecasting u. s. mortality[END_REF] modeled the logs of the age-specific death rates as a linear function of an unobserved period-specific intensity index, with parameters depending on age. We followed suit but had to translate the mortality rates by adding a constant (the smallest value of death rates of the 28 countries studied) to each death rate value before taking the natural logarithm. This step is crucial to avoid taking logarithms of zeros because the death rates for certain age groups from Luxembourg were zero. Each colour indicates a country in the log of death rates plots for males and females (Figures 2.2 Note: Data for the red shaded region is not available on HMD.

Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data

Methodology

Functional principal component analysis on areal spatial data

Consider n spatial locations i, one observed discrete measurement Y i,x,t taken at time t of location

i ∈ I ⊂ Z 2 , I a lattice region V , for a given x ∈ X = [0, T ] ⊂ N.
In our setting x is an age between 0 and T = 100 and Y i,x,t is a mortality rate observed at a year t ∈ D ⊂ N for a country of location i. We assume that for a given t these measurements points Y i,x,t are noisy observations of a smooth areal stochastic functional process {X i } i∈I :

Y i,x,t = µ t (x) + S i,t (x) + i,x,t = X i,t (x) + i,x,t (2.1)
where µ t is the mean function at time t (year). The n functions S i,t (.) are centered spatiotemporal squared integrable functional random variables on the space-time domain I ×D, namely S i (.) is valued in the Hilbert space L 2 (X ) endowed with the inner product f, g = X f (x)g(x)dx, for f, g in L 2 (X ). The unobserved variables { i,x,t , i = 1, ..., n} are i.i.d with zero mean Gaussian measurement errors and variance σ 2 .

We are interested in a functional PCA study where the classical PCA is replaced with its spatial counterpart, to consider spatial autocorrelation on the variable of interest in the sampling locations. This autocorrelation may be quantified by a weight matrix depending on the neighbour locations.

Let us consider in the following, the measurements of a given time t, ignoring the temporal distribution of different years. Let us fix the time t and delete the subscript t in equation ( 2.1), and consider S i,t (x) = S i (x) as a spatial functional variable and postulate a Karhunen-Loève expansion [START_REF] Ash | Topics in stochastic processes[END_REF]:

S i (x) = ∞ k=1 β k,i φ k (x), (2.2)
where φ k 's are the orthonormal eigenfunctions (functional principal components, FPC) and β k,i are auto-correlated scores. In practice, the sum is truncated to a finite integer, K which is to be chosen.

To compute the FPCs, let us express the sample data (S i ) i=1,...,n by means of a basis expansion:

S i (x) = ∞ m=1 c i,m B m (x) ≈ p m=1 c i,m B m (x), x ∈ X , (2.3)
where B m (.) is some collection of basis functions, c i,m = S i , B m have zero-mean. In practice, the first p functions are used where a sufficiently large p is good for approximation. [START_REF] Ramsay | Functional data analysis[END_REF] presented two main basis systems for building functions. The Fourier basis system is commonly used for periodic data while the B-spline basis system is preferable for nonperiodic data [START_REF] Ramsay | Functional data analysis with r and matlab: Springer science & business media[END_REF]. For the log death rates data, the B-spline basis system provides a more flexible basis. The selection of the number of basis functions is vital where a large number can lead to overfitting while a small number may cause underfitting.

The smoothing degree depends on the aim of the analysis. An important point we took note in setting the number of basis functions is the selection of a number less than the number of countries studied. The selection was made using step-by-step cross validation. In our case, we fit the data using 13 basis functions. The smoothing of the curves using B-spline was performed using the fda [START_REF] Ramsay | Fda: Functional data analysis [R package version 5[END_REF] package in the R software.

Estimation of the principal components and functional Moran's I statistic

Let us extend the well know Moran's statistic to the functional context. This aims to take into account the spatial dependency in the principal component analysis to analyze the degree of 20

Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data spatial autocorrelation among observations in the geographic space I [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF].

Let W = (W ij ) be a weighted spatial matrix where W ij is the neighbouring relation between locations i and j. Let W be standardised where the rows sum to one.

The functional Moran's index of the n row vector {S i (x)} i=1,...,n is introduced:

I n (S(x)) = n i=1 n j=1 W ij S i (x)S j (x) n i=1 S i (x) 2 = C n (S(x)) σ n (S(x)) , (2.4)
where

C n (S(x)) = 1 n n i=1 n j=1 W ij S i (x)S j (x) ≈ 1 n n i=1 n j=1 p m=1 p l=1 W ij c i,m c j,l B m (x)B l (y) = 1 n B(x) X W XB(x), (2.5)
and

σ n (S(x)) = 1 n n i=1 S i (x) 2 ≈ 1 n n i=1 p m=1 p l=1 c i,m c i,l B m (x)B l (y) = 1 n B(x) X XB(x).
(2.6)

X is the n × p matrix composed of the scores (c i,m ) i=1,...,n;m=1,...p of S i , B(x) is the p × 1 vector of components B m (x), m = 1, ..., p, S(x) is the p × 1 vector of functions S i (x).

The trace functional Moran's index is then introduced as:

I n (S) = T 0 I n (S(x))dx.
(2.7)

The classical univariate Moran's index [START_REF] Eckardt | Partial and semi-partial statistics of spatial associations for multivariate areal data[END_REF][START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]

of a n raw vector X m of components {c i,m } i=1,...,n is Ĩ(X m ) = X m W X m X m X m . Let V (X m ) = 1 n (X m X m ) Ĩ(X m ) = 1 n X m W X m .
It is highly positive when X m has a large variance and shows a global spatial structure and is negative in a situation with high variance and gives a local structure.

The purpose of the functional areal spatial principal component (FASPCA) proposed here is based on scaled R p vectors u (loadings) ( u =1) such that the n raw vectors χ = Xu are scattered and spatially autocorrelated. In other words, this aims to find the extreme values [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] of

C(u) = V (Xu) = 1 n u X W Xu.
(2.8)

The solutions [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] are the eigenvectors u k of 1 2n X (W +W )X associated with the largest and smallest eigenvalues α k = var(χ k ) Ĩ(χ k ) (where χ k = Xu k , var(χ k ) the variance of χ k ). Note that some eigenvalues α k may be negative since Ĩ(χ k ) is not always positive.

By the help of orthonormal vectors u k and their eigen-values α k , we introduce the estimated functional loading (eigen-function), φk (x) of the functional spatial areal PCA.

In fact, approximating X by

X ≈ X = K k=1 χ k u k ,
based on K (sufficiently large) relevant scores χ k corresponding to the K largest (in absolute values) eigen-values, lead to

S(x) ≈ XB(x) = K k=1 χ k u k B(x).
The functional spatial PCA is then obtained by letting the estimated eigen-functions as φk (x) = u k B(x) and the n-row functional scores βk = S (.) , φk (.) .

Then the FASPCA decomposition is obtained using equation (2.3) where the orthonormality of the vectors u k and the functions B m gives:

S i (x) ≈ K k=1 βk,i φk (x),
(2.9)

X i (x) ≈ μ(x) + K k=1 βk,i φk (x), (2.10)
where μ(x) = 1 n n i=1 X i (x), is the empirical mean with βk,i = T 0 S i (x) φk (x)dx.

Implemention of the functional Moran's I statistic on spatial weight matrices

This study aims at examining if spatial autocorrelation exists in the mortality rates for 28 European countries with data from HMD. The Moran's I statistic was extended and applied to the functional context. To ensure robustness, we used spatial weight matrices such as the knearest neighbour (KNN) and contiguity matrices. We constructed a KNN matrix of the 28 countries studied, belonging to the set of the five nearest neighbours of each other. For the contiguity weight matrix, we built a list of neighbours based on these 28 countries with contiguous boundaries where the single shared boundary point meets the contiguity condition. The weighted spatial matrix W ij can be classified into weights based on distance and weights based on boundaries. We use the distance d ij between each pair of spatial units (regions, cities, centroids, . . . ); i and j to construct spatial weight matrices for weights based on distance. The k-nearest neighbour weights are given as

w ij = 1 if j ∈ N k (i); 0 otherwise;
where N k (i) is the set of the k closest units or regions to i for k ∈ {1, . . . , n -1}. [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]. sPCA which complements PCA, was introduced to explicitly include spatial information in the analysis of genetic variation for investigating spatial genetic structures [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]. We employed FASPCA to reduce the dimensionality of data in our study since spatial information is a vital aspect in studying spatial autocorrelation. The PC scores of FASPCA consist of two types of patterns which are defined as global and local structures [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]. A global pattern differentiates between two spatial groups or a cline (or any intermediate state) while local scores retrieve stronger genetic differences among neighbours than among random pairs of entities [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]. The global pattern corresponds to positive spatial autocorrelation while the local pattern corresponds to negative spatial autocorrelation [START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF]. FASPCA was implemented using the fda [START_REF] Ramsay | Fda: Functional data analysis [R package version 5[END_REF], adegenet [START_REF] Jombart | Adegenet: A r package for the multivariate analysis of genetic markers[END_REF], ade4 [START_REF] Bougeard | Supervised multiblock analysis in R with the ade4 package[END_REF][START_REF] Chessel | The ade4 package -I: One-table methods[END_REF]Dray & Dufour, 2007;Dray et al., 2007) and adespatial [START_REF] Dray | Adespatial: Multivariate multiscale spatial analysis [R package version 0[END_REF] packages from the R software.

Results and discussion

The log death rates data for the 28 countries involving males and females described previously are analysed with the proposed FASPCA. First, the data are smoothed for convertion to functional objects. We then investigate the presence of spatial autocorrelation among the data. The log death rates data for the 28 countries involving males and females do not satisfy the normality assumption. We performed the Shapiro-Wilk test to assess multivariate normality (mvnormtest R package [START_REF] Jarek | Mvnormtest: Normality test for multivariate variables[END_REF]) on the log death rates data and found that this data violated the normality assumption. Hence, permutation tests for the Moran's I statistics are calculated for these data using 999 random permutations of the log death rates for all cases studied based on the KNN and contiguity weighting schemes. Table 2.1 shows the existence of significant spatial autocorrelation for female and male log death rates for years 1990, 2000 and 2010 for the 28 countries. In this , 2020). This contributed to the high spatial dependency (approximately 0.6) for these ages. We also observed that the spatial dependency increased from 1990 to 2010.

The formation of the European Union on 1 November 1993 could be the possible reason for the spatial dependence of mortality rates in these countries to increase over time as one of the objectives of the European Union is towards improving the quality of life of the population through cross-border cooperation, especially in healthcare.

We performed the classical FPCA to find new functions that reveal the most important types of variation in the curve data with the absence of the spatial structure. Table 2.2 gives the results for this FPCA on the male and female log death rates data for the aforesaid years and an autocorrelation Moran's test on the scores. The first PC reveals autocorrelation for both genders for the three years studied. Table 2.2 also reveals that the first PC alone accounts to more than 99.5% of the total variability of the data for all cases where we can see the data alone on one dimension. This does not give a clear picture of the data due to the absence of the spatial factor in the method.

We then proceeded to perform FASPCA for KNN and contiguity neighbours for the basis functions of the male and female log death rates data for all three years. Figure 2.5 gives a picture of the global structures (positive spatial autocorrelation, where the log of death rates are similar at neighbouring locations) and local structures (negative spatial autocorrelation, where the log of death rates tend to be dissimilar at neighbouring locations) to be retained for female and male data using KNN weights (Figure 2.6 for contiguity weights). Then, we ran the FASPCA described in Section 2.3 for the cases studied by considering the top three positive and top two negative eigen-values (Table 2.2). The percentage of variability explained by the functional principal components of FASPCA are given in Table 2.2.

The functional Moran's I statistics calculated based on these functional principal components show significant spatial autocorrelation for the principal components reported in Table 2.2 for KNN and contiguity weights (refer to p-values). Spatial autocorrelation can effectively be detected from the spatial principal components of the functional data which explains more than 95% of the percentage of variability based on the KNN and contiguity weights involving males and females. We reconstructed the data for each case by using the top two positive and top one negative PCs (Figures 2.2(c,d) and 2.3(c,d)). These PCs were mapped onto geographic spaces (representing 28 European countries generated using the maps package [START_REF] Becker | Enhancements by[END_REF] and the rgdal package [START_REF] Bivand | Rgdal: Bindings for the 'geospatial' data abstraction library[END_REF] where the black and white squares of the variable size represent positive and negative scores of the PCs respectively. The large black squares are well differentiated from the large white squares, while the small squares are less differentiated. The area of the square is proportional to the absolute value of the score. These graphical representations are applied to the significant PCs of each case for KNN and contiguity neighbours. The scores of the first PC from the classical FPCA were displayed on the geographical map (Figure 2.9). This FPCA does not consider spatial autocorrelation and as mentioned earlier, the first PC alone accounts to more than 99.5% of the total variability (Table 2.2) indicating its failure to cluster the locations based on spatial dependency. Hence, it is vital to consider the spatial aspect when performing PCA to identify spatially dependent locations on a geographical map. Since this is done on functional data, the functional Moran's I statistic will be the best tool to efficiently assess the spatial autocorrelation in functional data. We found that all the scores Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data belonged to the same group. This outcome was expected since the FPCA does not consider spatial autocorrelation and as mentioned earlier the first PC alone accounts to more than 99.5%

of the total variability (Table 2.2).

Concluding remarks

We extended the Moran's I statistic in the context of functional data analysis and applied it to the mortality data of 28 countries in Europe. The objective step in preparing the data to be used in the FDA context is by data smoothing. After converting the data into functions, spatial functional PCA was employed to find new functions that reveal the most important type of variation in the curve. The data was analysed from classical and spatial perspectives. Interestingly, we found the existence of spatial dependency in the mortality rates of neighbouring countries via the KNN and contiguity neighbourhood approaches. Our newly introduced functional Moran's I statistic is proved to be efficient in identifying the existence of spatial dependency of mortality rates of neighbouring countries effectively using functional principal components, which are outcomes of dimensionality reduction. In our future work, this idea of spatial dependency Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data in mortality rates will be extended towards constructing a spatial predictive model to predict mortality rates for neighbouring countries with limited or no data. It is also of interest to further modify this work to the spatio-temporal framework.

Appendix

Introduction

Agriculture, economics, environmental sciences, urban systems, and epidemiology activities often utilize spatially dependent data. Therefore, modelling such activities requires one to find a type of correlation between some random variables in one location with other variables in neighbouring locations; see for instance [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. This is a significant feature of spatial data analysis. Spatial/Econometrics statistics provides tools to perform such modelling. Many studies on spatial effects in statistics and econometrics using many diverse models have been published;

see [START_REF] Cressie | Statistics for spatial data[END_REF], [START_REF] Anselin | Thirty years of spatial econometrics[END_REF], [START_REF] Anselin | Spatial econometrics: Methods and models[END_REF] and [START_REF] Arbia | Spatial econometrics: Statistical foundations and applications to regional convergence[END_REF] for a review.

Two main methods of incorporating a spatially dependent structure (see for instance [START_REF] Cressie | Statistics for spatial data[END_REF] can essentially be distinguished as between geostatistics and lattice data. In the domain of geostatistics, the spatial location is valued in a continuous set of R N , N ≥ 2. However, for many activities, the spatial index or location does not vary continuously and may be of the lattice type, the baseline of this current work. In image analysis, remote sensing from satellites, agriculture etc., data are often received as a regular lattice and identified as the centroids of square pixels, whereas a mapping often forms an irregular lattice. Basically, statistical models for lattice data Chapter 3. Partially linear spatial probit models are linked to nearest neighbours to express the fact that data are nearby.

Two popular spatial dependence models have received substantial attention for lattice data, the spatial autoregressive (SAR) dependent variable model and the spatial autoregressive error model (SAE, where the model error is an SAR), which extend the regression in a time series setting to spatial one.

From a theoretical point of view, various linear spatial regression SAR and SAE models as well as their identification and estimation methods, e.g., two-stage least squares (2SLS), three-stage least squares (3SLS), maximum likelihood (ML) or quasi-maximum likelihood (QML) and the generalized method of moments (GMM), have been developed and summarized by many authors such as [START_REF] Anselin | Spatial econometrics: Methods and models[END_REF], [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF], [START_REF] Kelejian | A generalized moments estimator for the autoregressive parameter in a spatial model[END_REF], [START_REF] Conley | Gmm estimation with cross sectional dependence[END_REF], [START_REF] Cressie | Statistics for spatial data[END_REF], [START_REF] Case | Spatial patterns in household demand[END_REF], L.-F. [START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF], L.-f. [START_REF] Lee | GMM and 2SLS estimation of mixed regressive, spatial autoregressive models[END_REF], [START_REF] Lin | GMM estimation of spatial autoregressive models with unknown heteroskedasticity[END_REF], [START_REF] Zheng | On the asymptotics of maximum likelihood estimation for spatial linear models on a lattice[END_REF], [START_REF] Malikov | Semiparametric estimation and testing of smooth coefficient spatial autoregressive models[END_REF], [START_REF] Garthoff | Control charts for multivariate spatial autoregressive models[END_REF], [START_REF] Yang | Identification and QML estimation of multivariate and simultaneous equations spatial autoregressive models[END_REF].

Introducing nonlinearity into the field of spatial linear lattice models has attracted less attention; see [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF], who generalised kernel regression estimation to spatial lattice data.

Su ( 2012) proposed a semi-parametric GMM estimation for some semi-parametric SAR models.

Extending these models and methods to discrete choice spatial models has seen less attention where only a few researches are concerned with this topic in recent years. This may be as noted by [START_REF] Fleming | Techniques for estimating spatially dependent discrete choice models[END_REF] (see also [START_REF] Smirnov | Modeling spatial discrete choice[END_REF] and [START_REF] Billé | Computational issues in the estimation of the spatial probit model: A comparison of various estimators[END_REF]) due to the "added complexity that spatial dependence introduces into discrete choice models". Estimating the model parameters with a full ML approach in spatially discrete choice models often requires solving a very computationally demanding problem of n-dimensional integration, where n is the sample size.

For linear models, many discrete choice models are fully linear and utilize a continuous latent variable; see [START_REF] Smirnov | Modeling spatial discrete choice[END_REF], H. [START_REF] Wang | Partial maximum likelihood estimation of spatial probit models[END_REF] and [START_REF] Martinetti | Approximate likelihood estimation of spatial probit models[END_REF], who proposed pseudo-ML methods, and Pinkse and Slade (1998) who studied a method based on the GMM approach. Also, others methodologies of estimation are used such as the EM algorithm [START_REF] Mcmillen | Probit with spatial autocorrelation[END_REF] and Gibbs sampling approach [START_REF] Lesage | Bayesian estimation of limited dependent variable spatial autoregressive models[END_REF].

When the relationship between the discrete choice variable and some explanatory variables is not linear, a semi-parametric model may be an alternative to fully parametric models. This type of model is known in the literature as partially linear choice spatial models and is the baseline of this current work. When the data are independent, these choice models can be viewed as special cases of the famous generalised additive models [START_REF] Hastie | Generalized additive models[END_REF] and have received substantial attention in the literature, and various estimation methods have been explored (see for instance [START_REF] Carroll | Generalized partially linear single-index models[END_REF][START_REF] Hunsberger | Semiparametric regression in likelihood-based models[END_REF][START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF].

To the best of our knowledge, semi-parametric spatial choice models have not yet been investigated from a theoretical point of view. To fill this gap, this work addresses an SAE spatial probit model for when the spatial dependence structure is integrated in a disturbance term of the studied model.

We propose a semi-parametric estimation method combining the GMM approach and the weighted likelihood method. The method consists of first fixing the parametric components of the model and non-parametrically estimating the non-linear component by weighted likelihood [START_REF] Staniswalis | The kernel estimate of a regression function in likelihood-based models[END_REF]. The obtained estimator depending on the values at which the parametric components are fixed is used to construct a GMM estimator [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]) of these components.

The remainder of this paper is organised as follows. In Section 3.1, we introduce the studied spatial model and the estimation procedure. Section 3.2 is devoted to hypotheses and asymptotic results, while Section 3.3 reports a discussion and computation of the estimates. Section 3.4 gives

3.1. Model some numerical results based on simulated data to illustrate the performance of the proposed estimators. The last section presents the proofs of the main results.

Model

We consider that at n spatial locations {s 1 , s 2 , . . . , s n } satisfying s i -s j > ρ with ρ > 0, observations of a random vector (Y, X, Z) are available. Assume that these observations are considered as triangular arrays [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF] and follow the partially linear model of a latent dependent variable Y * :

Y * in = X T in β 0 + g 0 (Z in ) + U in , 1 ≤ i ≤ n, n = 1, 2, . . . (3.1)
with In model (3.1), β 0 and g 0 (•) are constant over i (and n). Assume that the disturbance term U in in (3.2) is modelled by the following spatial autoregressive process (SAR):

Y in = I (Y * in ≥ 0) , 1 ≤ i ≤ n, n = 1, 2, . . . ( 3
U in = λ 0 n j=1 w ijn U jn + ε in , 1 ≤ i ≤ n, n = 1, 2, . . . (3.3)
where, we assume that, for all n = 1, 2, . . ., {ε in ,

1 ≤ i ≤ n} is independent of {X in , 1 ≤ i ≤ n} and {Z in , 1 ≤ i ≤ n}, and {X in , 1 ≤ i ≤ n} is independent of {Z in , 1 ≤ i ≤ n}.
λ 0 is the autoregressive parameter, valued in the compact subset Θ λ ⊂ R, w ijn , j = 1, ..., n are the elements in the i-th row of a non-stochastic n × n spatial weight matrix W n , which contains the information on the spatial relationship between observations. This spatial weight matrix is usually constructed as a function of the distances (with respect to some metric) between locations;

see [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF] for additional details. The n × n matrix (I n -λ 0 W n ) is assumed to be non-singular for all n, where I n denotes the n × n identity matrix and {ε in , 1 ≤ i ≤ n} are assumed to be independent random Gaussian variables; E(ε in ) = 0 and E(ε 2 in ) = 1 for i = 1, . . . , n n = 1, 2, . . .. Note that one can rewrite (3.3) as .4) where U n = (U n1 , . . . , U nn ) T and ε n = (ε n1 , . . . , ε nn ) T . Therefore, the variance-covariance matrix of U n is .5) This matrix allows one to describe the cross-sectional spatial dependencies between the n observations. Furthermore, the fact that the diagonal elements of V n (λ 0 ) depend on λ 0 and particularly on i and n allows some spatial heteroscedasticity. These spatial dependencies and heteroscedasticity depend on the neighbourhood structure established by the spatial weight ma-trix W n .

U n = (I n -λ 0 W n ) -1 ε n , n = 1, 2, . . . ( 3
V n (λ 0 ) ≡ Var(U n ) = (I n -λ 0 W n ) -1 (I n -λ 0 W n ) T -1 , n = 1, 2, . . . ( 3
The elements w ijn of W n are usually considered as inversely proportional to the distance between spatial units i and j with respect to some metric Pinkse and Slade (physical distance, social network or economic distance, see for instance 1998). The matrices W n are usually classified into two groups: Weights Based on Distance and Weights Based on Boundaries. For Weights Based on Distance, the distance d ij between each pair of spatial units (regions, cities, centroids,...) i and j are basically considered.

• k-Nearest Neighbor weights

w ij = 1 if j ∈ N k (i),
0 Otherwise where N k (i) is the set of the k closest units or regions to i for k ∈ {1, ..., n -1}

• Power Distance Decay weights

w ij = d -α ij if 0 ≤ d ij ≤ δ, 0 if d ij > δ where α is any positive exponent, typically α = 1 or α = 2.
For Weights Based on Boundaries, spatial contiguity is often used to specify neighboring location in the sense of sharing a common border. There are different type of spatial contiguity but the classical cases are those referred to Rook contiguity (with only common boundaries), Bishop contiguity (with only common vertices) and Queen contiguity (with both Rook and Bishop contiguity).

w ij =
1 if i and j are contiguity 0 Otherwise In general, we can rewrite the last equation as:

w ij = 1 ij > 0 0 ij = 0 ,
with ij denotes the length of shared boundary.

Before proceeding further, let us give some particular cases of the model.

If one considers i.i.d observations, that is, V n (λ 0 ) = σ 2 I n , with σ depending on λ 0 , the obtained model may be viewed as a special case of classical generalised partially linear models (e.g. [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF] or the classical generalised additive model [START_REF] Hastie | Generalized additive models[END_REF].

Several approaches for estimating this particular model have been developed; among these methods, we cite that of [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF] based on the concept of the generalised profile likelihood Severini and Wong (e.g 1992). This approach consists of first fixing the parametric parameter β and non-parametrically estimating g 0 (•) using the weighted likelihood method. This last estimate is then used to construct a profile likelihood to estimate β 0 .

When g 0 ≡ 0 (or is an affine function), that is, without a non-parametric component, several approaches have been developed to estimate the parameters β 0 and λ 0 . The basic difficulty encountered is that the likelihood function of this model involves an n-dimensional normal integral; thus, when n is high, the computation or asymptotic properties of the estimates may present difficulties (e.g. [START_REF] Poirier | Probit with dependent observations[END_REF]. Various approaches have been proposed to addressed this difficulty; among these approaches, we cite the following:

• Feasible Maximum Likelihood approach: this approach consists of replacing the true likelihood function by a pseudo-likelihood function constructed via marginal likelihood functions. Smirnov ( 2010) proposed a pseudo-likelihood function obtained by replacing V n (λ 0 )

3.1. Model

35

by some diagonal matrix with the diagonal elements of V n (λ 0 ). Alternatively, H. [START_REF] Wang | Partial maximum likelihood estimation of spatial probit models[END_REF] proposed to divide the observations by pairwise groups, where the latter are assumed to be independent with a bivariate normal distribution in each group, and estimate β 0 and λ 0 by maximizing the likelihood of these groups. Recently [START_REF] Martinetti | Approximate likelihood estimation of spatial probit models[END_REF] proposed a pseudo-likelihood function defined as an approximation of the likelihood function where the latter is inspired by some univariate conditioning procedure.

• Generalised Method of Moments (GMM) approach used by [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. These authors used the generalized residuals defined by Ũin (β, λ) = E (U in |Y in , β, λ) , 1 ≤ i ≤ n, n = 1, 2, . . . with some instrumental variables to construct moment equations to define the GMM estimators of β 0 and λ 0 .

In what follows, using the n observations (X in , Y in , Z in ), i = 1, ..., n, we propose parametric estimators of β 0 , λ 0 and a non-parametric estimator of the smooth function g 0 (•).

We give asymptotic results according to increasing domain asymptotic. This consists of a sampling structure whereby new observations are added at the edges (boundary points) compare to the infill asymptotic, which consists of a sampling structure whereby new observations are added in-between existing observations. A typical example of an increasing domain is lattice data. An infill asymptotic is appropriate when the spatial locations are in a bounded domain.

Estimation Procedure

We propose an estimation procedure based on a combination of a weighted likelihood method and a generalized method of moments. We first fix the parametric components β and λ of the model and estimate the non-parametric component using a weighted likelihood. The obtained estimate is then used to construct generalised residuals, where the latter are combined with the instrumental variables to propose GMM parametric estimates. This approach will be described as follows:

By equation (3.2), we have

E 0 (Y in |X in , Z in ) = Φ (v in (λ 0 )) -1 X T in β 0 + g 0 (Z in ) , 1 ≤ i ≤ n, n = 1, 2, . . . (3.6)
where E 0 denotes the expectation under the true parameters (i.e., β 0 , λ 0 and g 0 (•)), Φ(•) is the cumulative distribution function of a standard normal distribution, and

(v in (λ 0 )) 2 = V iin (λ 0 ), 1 ≤ i ≤ n, n = 1, 2, • • • are the diagonal elements of V n (λ 0 ). For each β ∈ Θ β , λ ∈ Θ λ , z ∈ Z and η ∈ R, we define the conditional expectation on Z in of the log-likelihood of Y in for 1 ≤ i ≤ n, n = 1, 2, . . ., as H(η; β, λ, z) = E 0 L Φ (v in (λ)) -1 η + X T in β ; Y in Z in = z , (3.7) with L(u; v) = log u v (1 -u) 1-v .
Note that H(η; β, λ, z) is assumed to be constant over i (and n). For each fixed β ∈ Θ β , λ ∈ Θ λ and z ∈ Z, g β,λ (z) denotes the solution in η of

∂ ∂η H(η; β, λ, z) = 0. (3.8)
Then, we have g β 0 ,λ 0 (z) = g 0 (z) for all z ∈ Z.

Chapter 3. Partially linear spatial probit models

Now, using g β,λ (•), we construct the GMM estimates of β 0 and λ 0 as in [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF] For that, we define the generalised residuals, replacing g 0 (Z in ) in (3.1) by g β,λ (Z in ):

Ũin (β, λ, g β,λ ) = E (U in |Y in , β, λ) (3.9) = φ (G in (β, λ, g β,λ )) (Y in -Φ (G in (β, λ, g β,λ ))) Φ (G in (β, λ, g β,λ )) (1 -Φ (G in (β, λ, g β,λ ))) ,
where φ(•) is the density of the standard normal distribution and

G in (β, λ, g β,λ ) = (v ni (λ)) -1 X T in β + g β,λ (Z in ) .
For notational simplicity, we write θ = (β T , λ) T ∈ Θ ≡ Θ β × Θ λ when possible.

Note that in (3.9), the generalised residual Ũin (• , •) is calculated by conditioning only on Y in and not on the entire sample {Y in , i = 1, 2, . . . , n, n = 1, . . .} or a subset of it. This of course will influence the efficiency of the estimators of θ obtained by these generalised residuals, but it allows one to avoid a complex computation; see [START_REF] Poirier | Probit with dependent observations[END_REF] for additional details.

To address this loss of efficiency, let us follow Pinkse and Slade (1998)'s procedure, which consists of employing some instrumental variables to create some moment conditions, and use a random matrix to define a criterion function. Both the instrumental variables and the random matrix permit one to consider more information about the spatial dependences and heteroscedasticity characterizing the dataset. Let us now discuss the details of the estimation procedure. Let

S n (θ, g θ ) = n -1 ξ T n Ũn (θ, g θ ), (3.10)
where Ũn (θ, g θ ) is an n × 1 vector, composed of Ũin (θ, g θ ), 1 ≤ i ≤ n and ξ n is an n × q matrix of instrumental variables, whose ith row is given by the 1 × q random vector ξ in . The latter may depend on g θ (•) and θ. We assume that ξ in is σ(X in , Z in ), measurable for each i = 1, . . . , n, n = 1, 2, . . .. We suppress the possible dependence of the instrumental variables on the parameters for notational simplicity. The GMM approach consists of minimising the following sample criterion function:

Q n (θ, g θ ) = S T n (θ, g θ )M n S n (θ, g θ ), (3.11)
where M n is some positive-definite q × q weight matrix that may depend on the sample information. The choice of the instrumental variables and weight matrix characterizes the difference between GMM estimator and all pseudo-maximum likelihood estimators. For instance, if one takes

ξ in (θ, g θ ) = ∂G in (θ, η i ) ∂θ + ∂G in (θ, η i ) ∂η ∂g θ ∂θ (Z in ), (3.12
)

with η i = g θ (Z in ), G in (θ, η i ) = (v in (λ)) -1 X T in β + η i
, and M n = I q with q = p + 1, then the GMM estimator of θ is equal to a pseudo-maximum profile likelihood estimator of θ, accounting only for the spatial heteroscedasticity.

Now, let

S(θ, g

θ ) = lim n→∞ E 0 (S n (θ, g θ )) , (3.13)
and

Q(θ, g θ ) = S T (θ, g θ )M S(θ, g θ ),
where M , the limit of the sequence M n , is a nonrandom positive-definite matrix. The functions

S n (•, •) and Q n (•, •) are viewed as empirical counterparts of S(•, •) and Q(•, •), respectively.
Clearly, g θ (•) is not available in practice. However, we need to estimate it, particularly by an asymptotically efficient estimate. By (3.8) and for fixed θ T = (β T , λ) ∈ Θ, an estimator of g θ (z),

for z ∈ Z, can be given by ĝθ (z), which denotes the solution in η of or more generally the local polynomial method [START_REF] Fan | Local polynomial modelling and its applications: Monographs on statistics and applied probability[END_REF], can be used to reduce this bias. Another alternative is to use trimming [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF], in which the function S n (θ, g θ ) is computed using only observations associated with Z in that are away from the boundary. The advantage of this approach is that the theoretical results can be presented in a clear form, but it is less tractable from a practical point of view, in particular, for small sample sizes.

n i=1 ∂ ∂η L (Φ (G in (θ, η)) ; Y in ) K z -Z in b n = 0, ( 3

Large sample properties

We now turn to the asymptotic properties of the estimators derived in the previous section: θT = ( βT , λ) and ĝθ (•). Let us use the following notation: d dθ S(θ, g θ ) means that we differentiate S(., .) with respect to θ, and ∂ ∂θ S(θ, g θ ) is the partial derivative of S(•, •) w.r.t the first variable. The partial derivative of S n (θ, g) w.r.t g, for any function v ∈ G, is

∂S n ∂g (θ, g)(v) = n -1 n i=1 ξ in ∂ Ũin ∂η (θ, η i )v(Z in ).
Without ambiguity, a denotes sup t |a(t)| when a is a function, a 2 i 1/2 when a is a vector, and

a 2 ij 1/2
when a is a matrix.

Let the following matrices be needed in the asymptotic variance-covariance matrix of θ:

B 1 (θ 0 ) = lim n→∞ E 0 nS n (θ 0 , g 0 ) S T n (θ 0 , g 0 ) , B 2 (θ 0 ) = d dθ S T (θ, g θ ) θ=θ 0 M d dθ S (θ, g θ ) θ=θ 0 , with d dθ S (θ, g θ ) = ∂S ∂θ (θ, g θ ) + ∂S ∂g (θ, g θ ) ∂ ∂θ g θ , (3.16)
and

Ω(θ 0 ) = {B 2 (θ 0 )} -1 d dθ S T (θ, g θ ) θ=θ 0 M B 1 (θ 0 )M d dθ S (θ, g θ ) θ=θ 0 {B 2 (θ 0 )} -1 .
Chapter 3. Partially linear spatial probit models

The following assumptions are required to establish the asymptotic results.

Assumption A1. (Smoothing condition).

For each fixed θ ∈ Θ and z ∈ Z, let g θ (z) denote the unique solution with respect to η of ∂ ∂η H(η; θ, z) = 0.

For any ε > 0 and g ∈ G, there exists γ > 0 such that f in (z) > 0.

sup θ∈Θ,z∈Z ∂ ∂η H(g(z); θ, z) ≤ γ =⇒ sup θ∈Θ,z∈Z |g(z) -g θ (z)| ≤ ε. ( 3
(3.18)

The joint probability density f ijn (., .) of (Z in , Z jn ) exists and is bounded on Z × Z uniformly on i = j and n.

Assumption A3. (Spatial dependence). Let h θ, η i in (•|•, •) denote the conditional log likelihood function of Y in given (X in , Z in )
, where η i = g(Z in ). Let T in be the vector (Y in , X in , Z in ), i = 1, . . . , n , n = 1, 2 . . ., p = p + 1, and assume that for all i, l = 1, . . . , n,

|Cov 0 (ψ(T in ), ψ(T ln ))| ≤ {Var 0 (ψ(T in )) Var 0 (ψ(T ln ))} 1/2 α iln , (3.19)
with

ψ(T in ) = K z -Z in b n or ψ(T in ) = K z -Z in b n ∂ j 1 +•••+j p+r ∂θ j 1 1 • • • ∂θ j p p ∂η r h θ, η in (Y in | X in , Z in = z),
for all z ∈ Z, θ ∈ Θ, η = g(z) with g ∈ G, and for all nonnegative integers j 1 , . . . , j p = 0, 1, 2 and r = 0, . . . , 4, such that j 1 + • • • + j p + r ≤ 6.

We assume that

Cov 0 ξ itn Ũin (θ, g θ ), ξ jsn Ũjn (θ, g θ ) ≤ Var 0 ξ itn Ũin (θ, g θ ) Var 0 ξ jsn Ũjn (θ, g θ )

1/2 α ijn , (3.20)
for all θ ∈ Θ, i, j = 1, . . . , n, n = 1, 2, . . . and for any s, t = 1, . . . , q, and Cov 0 ξ

(2)

in (θ 0 , η 0 i ), ξ (2)
jn (θ 0 , η 0 j ) ≤ Var 0 ξ (2) in (θ 0 , η 0 i ) Var 0 ξ (2) jn (θ 0 , η 0 j ) 1/2 α ijn , (3.21) with ξ (2) in (θ 0 , η 0 i ) := w T ξ i Λ G in (θ 0 , η 0 i ) φ G in (θ 0 , η 0 i ) ∂G in ∂θ (θ 0 , η 0 i ),
where η 0 i = g 0 (Z i ) for each w ∈ R q such that w = 1. In addition, assume that there is a decreasing (to 0) positive function ϕ(•) such that the "mixing" numbers verify α ijn = O (ϕ ( s i -s j )), r 2 ϕ(rr * )/ϕ(r * ) = o(1), as r → 0, for all fixed r * > 0, where s i and s j are spatial coordinates associated with observations i and j, respectively.

Assumption A4. The kernel K satisfies K(u)du = 1. It is Lipschitzian, i.e., there is a positive constant C such that |K(u) -K(v)| ≤ C u -v for all u, v ∈ R d . Assumption A5. The bandwidth b n satisfies b n → 0 and nb 3d+1 n → ∞ as n → ∞.
Assumption A6. The instrumental variables satisfy sup i, n ξ in = O p (1), where ξ in is the i-th column of the n × q matrix of instrumental variables ξ n .

Assumption A7. θ T = (β T , λ) takes values in a compact and convex set Θ = Θ β × Θ λ ⊂ R p × R, and θ T 0 = (β T 0 , λ 0 ) is in the interior of Θ. Assumption A8. S(•, •) is continuous on both arguments θ and g, and Q(•, g . ) attains a unique minimum over Θ at θ 0 .

Assumption A9. The square root of the diagonal elements of V n (λ) are twice continuous differentiable functions with respect to λ and sup

λ∈Θ λ v -1 in (λ) + d dλ v in (λ) + d 2 dλ 2 v in (λ) < ∞ uniformly on i and n.
Assumption A10. B 1 (θ 0 ) and B 2 (θ 0 ) are positive-definite matrices, and M n -M = o p (1).

Remark 1. Assumption A1 ensures the smoothness of H(.; ., .) around its extrema point g θ (.); see [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF]. Assumption A2 is a decay of the local independence condition of the covariates Z in , meaning that these variables are not identically distributed; a similar condition can be find in [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF]. Condition (3.18) generalizes the classical assumption inf z f (z) > 0 used in the case of estimating the density function f (•) with identically distributed or stationary random variables. This assumption has been used in [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF] (Assumption A7(x), p. 8). Assumption A3 describes the spatial dependence structure, it is a particular case of the Assumption A in [START_REF] Pinkse | A central limit theorem for endogenous locations and complex spatial interactions[END_REF] and may be verified by mixing random variables, see [START_REF] Pinkse | A central limit theorem for endogenous locations and complex spatial interactions[END_REF] for more details. Note that the processes that we use are not assumed stationary; this allows for greater generalizability and the dependence structure to change with the sample size n (see [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF] for more discussion). Conditions (3.19), (3.20) and (3.21) are not restrictive. When the regressors and instrumental variables are deterministic, conditions (3.19) and (3.20) are equivalent to |Cov 0 (Y in , Y ln )| ≤ α iln . The condition on ϕ(•) is satisfied when the latter tends to zero at a polynomial rate, i.e., ϕ(t) = O(t -τ ), for all τ > 2, as in the case of mixing random variables.

Assumption A6 requires that the instruments and explanatory variables be bounded uniformly on i and n. In addition, when the instruments depend on θ and g(•), they are also uniformly bounded with respect to these parameters. The compactness condition in Assumption A7 is standard, and the convexity is somewhat unusual; however, it is reasonable in most applications. Condition A8 is necessary to ensure the identification of the true parameters θ 0 . Assumption A9 requires the standard deviations of the errors to be uniformly bounded away from zero with bounded derivatives. This has been considered by [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]. Assumption A10 is classic [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF]) and required in the proof of Theorem 3.2.2. Those authors noted that in their model (without a non-parametric component), when the autoregressive parameter λ 0 = 0, B 2 (θ 0 ) is not invertible, regardless of the choice of M n . This is also the case in our context because for each g θ (z) solution of (3.8), θ ∈ Θ and z ∈ Z, we have

∂g θ ∂β (z) = - E ( Γ jn (θ, g θ (z))X jn | Z jn = z) E ( Γ jn (θ, g θ (z))| Z jn = z) ,
and

∂g θ ∂λ (z) = v jn (λ) v jn (λ) E Γ jn (θ, g θ (z)) X T jn β + g θ (z) Z jn = z E ( Γ jn (θ, g θ (z))| Z jn = z) = v jn (λ) v jn (λ) g θ (z) -β T ∂g θ ∂β (z) , where v jn (λ) = d dλ v jn (λ) = v jn (λ) W n S -1 n (λ)V n (λ) jj , Γ jn (•) = Λ (G jn (•)) [Y jn -Φ(G jn (•))] -Λ (G jn (•)) φ (G jn (•))
and

Λ(•) = φ(•)/(1 -Φ(•))Φ(•). However ∂g θ ∂λ (z) λ=0 = 0 because v jn (0) = 0,
then B 2 (θ 0 ) will be singular when λ 0 = 0.

With these assumptions in place, we are able to give some asymptotic results. The weak consistencies of the proposed estimators are given in the following two results. The first theorem and corollary below establish the consistency of our estimators, whereas the second theorem addresses the question of convergence to a normal distribution of the parametric component when it is properly standardised. Proof of Corollary 3.2.1 Note that

ĝθ -g 0 ≤ ĝθ -g θ + g θ -g 0 ≤ sup θ ĝθ -g θ + sup θ ∂g θ ∂θ θ -θ 0 = o p (1),
since, by the assumptions of Theorem 3.2.1, sup θ ĝθ -g θ = o p (1) and sup θ ∂g θ ∂θ < ∞.

The following gives an asymptotic normality result of θ.

Theorem 3.2.2. Under assumptions A1-A10, we have

√ n θ -θ 0 → N (0, Ω(θ 0 )) Remark 2.
In practice, the previous asymptotic normality result can be used to construct asymptotic confidence intervals and build hypothesis tests when a consistent estimate of the asymptotic covariance matrix Ω(θ 0 ) is available. To estimate this matrix, let us follow the idea of Pinkse and Slade (1998) and define the estimator

Ω n ( θ) = B 2n ( θ) -1 d dθ S T n (θ, ĝθ ) θ= θ M n B 1n ( θ)M n d dθ S n (θ, ĝθ ) θ= θ B 2n ( θ) -1 , with B 1n (θ) = nS n (θ, ĝθ )S T n (θ, ĝθ ) and B 2n (θ) = d dθ S T n (θ, ĝθ ) M n d dθ S n (θ, ĝθ ) .
The consistency of Ω n ( θ) will be based on that of B 1n ( θ) and B 2n ( θ), the estimators of B 1 (θ 0 )

and B 2 (θ 0 ), respectively. Note that the consistency of B 2n ( θ) is relatively easy to establish. On the other hand, that of B 1n ( θ) asks for additional assumptions and an adaption of the proof of Theorem 3 of Pinkse and Slade (1998, p.134) to our case; this is of interest to future research.

Computation of the estimates

The aim of this section is to outline in detail how the regression parameters β, the spatial autocorrelation parameter λ and the non-linear function g θ can be estimated. We begin with the computation of ĝθ (z), which will play a crucial role in what follows.

Computation of the estimate of the non-parametric component

An iterative method is needed to compute the ĝθ (z) solution of (3.14) for each fixed θ ∈ Θ and z ∈ Z. For fixed θ T = (β, λ) ∈ Θ and z ∈ Z, let η θ = g θ (z) and ψ(η; θ, z) denote the left-hand side of (3.14), which can be rewritten as

ψ(η; θ, z) = n i=1 [v in (λ)] -1 Λ (G in (θ, η)) [Y in -Φ(G in (θ, η))] K z -Z in b n . (3.22)
Consider the Fisher information:

Ψ (η θ ; θ, z) = E 0 ∂ ∂η ψ(η; θ, z) η=η θ {(X in , Z in ), 1 ≤ i ≤ n, n = 1, . . .} = - n i=1 [v in (λ)] -2 Λ (G in (θ, η θ )) φ (G in (θ, η θ )) K z -Z in b n + (3.23) n i=1 [v in (λ)] -2 Λ (G in (θ, η θ )) [Φ (G in (θ 0 , η 0 )) -Φ (G in (θ, η θ ))] K z -Z in b n
Note that the second term in the RHS (Right Hand Side) of (3.24) is negligible when θ is near the true parameter θ 0 .

Because ψ(η; θ, z) = 0 for η = ĝθ (z), an initial estimate η can be updated to η † using Fisher's scoring method:

η † = η - ψ(η; θ, z) Ψ (η; θ, z) . (3.24)
The iteration procedure (3.24) requests some starting value η = η0 to ensure convergence of the algorithm. To this end, let us adapt the approach of [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF], which

Chapter 3. Partially linear spatial probit models consists of supposing that for fixed θ ∈ Θ, there exists a η0 satisfying G in (θ, η0

) = Φ -1 (Y in ) for i = 1, . . . , n. Knowing that G in (θ, η0 ) = (v in (λ)) -1 X T ni β + η0 , we have η0 = v in (λ)Φ -1 (Y in ) - X T in β.
Then, (3.24) can be updated using the following initial value:

η † 0 = η0 - ψ( η0 ; θ, z) Ψ ( η0 ; θ, z) = n i=1 [v in (λ)] -1 Λ(C in )φ(C in ) C in -[v in (λ)] -1 X T in β K z-Z in bn n i=1 [v in (λ)] -2 Λ(C in )φ(C in )K z-Z in bn ,
where

C in = Φ -1 (Y in ), i = 1, . . . , n, is computed using a slight adjustment because Y in ∈ {0, 1}.
With this initial value, the algorithm iterates until convergence.

Selection of the bandwidth

A critical step (in non-or semi-parametric models) is the choice of the bandwidth parameter b n , which is usually selected by applying some cross-validation approach. The latter was adapted by [START_REF] Su | Semiparametric gmm estimation of spatial autoregressive models[END_REF] in the case of a spatial semi-parametric model. Because cross-validation may be very time consuming, which is true in the case of our model, we adapt the following approach used in [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF] to achieve greater flexibility:

1. Consider the linear regression of C in on X in , i = 1, . . . , n, without an intercept term, and let R 1n , . . . , R nn denote the corresponding residuals.

2. Since we expect E(R in |Z in = z) to have similar smoothness properties as g 0 (.), the optimal bandwidth b n is that of the non-parametric regression of the

{R in } i=1,••• ,n on {Z in } i=1,••• ,n ,
chosen by applying any non-parametric regression bandwidth selection method. For that, we use the cross-validation method in the np R Package.

Computation of θ

The parametric component β and the spatial autoregressive parameter λ are computed as mentioned above by a GMM approach based on some instrumental variables ξ n and the weight matrix M n . The choices of these instrumental variables and weight matrix M n are as follows.

Because ψ(ĝ θ (z); θ, z) = 0, if we differentiate the latter with respect to β and λ, we have

∂ ∂β ĝθ (z) = - n i=1 [v in (λ)] -2 ∆ in (θ, z)X in K z-Z in bn n i=1 [v in (λ)] -2 ∆ in (θ, z)K z-Z in bn , and
∂ ∂λ ĝθ (z) = n i=1 [v in (λ)] -1 v in (λ)∆ in (θ, z) X T in β + ĝθ (z) K z-Z in bn n i=1 [v in (λ)] -2 ∆ in (θ, z)K z-Z in bn + n i=1 [v in (λ)] -2 v in (λ)Λ (G in (θ, ĝθ (z))) [Y in -Φ (G in (θ, ĝθ (z)))] K z-Z in bn n i=1 [v in (λ)] -2 ∆ in (θ, z)K z-Z in bn , with ∆ in (θ, z) = Λ (G in (θ, ĝθ (z))) [Y in -Φ (G in (θ, ĝθ (z)))] -Λ (G ni (θ, ĝθ (z))) φ (G in (θ, ĝθ (z))) .
Then, the previous result is used to define the following instrumental variables:

ξ in (θ, ĝθ ) = ∂G in (θ, ηi ) ∂θ + ∂G in (θ, ηi ) ∂η ∂ ∂θ ĝθ (Z in ),
with ηi = ĝθ (Z in ).

For the weight matrix, one can use M n = I q with q = p + 1 as in [START_REF] Pinkse | Contracting in space: An application of spatial statistics to discrete-choice models[END_REF].

Then, the obtained GMM estimator of θ with this choice of M n is equal to the pseudo-profile maximum likelihood estimator of θ, accounting only for the spatial heteroscedasticity. Another empirical choice could be the idea of continuous updating GMM estimator (One step GMM) used in [START_REF] Pinkse | Dynamic spatial discrete choice using one-step gmm: An application to mine operating decisions[END_REF]:

M n (θ) =    n -1 n i,j=1 δ ij ξ ni ξ T jn Ũin (θ, ĝθ ) Ũjn (θ, ĝθ )    -1 (3.25)
with the weights

δ ij = n r=1 τ ri τ rj n r=1 τ 2 ri n r=1 τ 2 rj 1/2 for i, j = 1, . . . , n,
where τ ij is a number depending on w nij such that the nearer location i is to location j, the larger τ ij is. For instance, we expect to have more efficient estimators with this matrix.

Finite sample properties

In this section, we study the performance of the proposed model based on some numerical results, which highlight the importance of accounting for both the spatial dependence and the partial linearity. Random datasets from the following spatial semi-parametric models are generated and first we investigate the estimation quality of the proposed procedure which accounts both the spatial dependence and the partial linearity. The influences of the spatial dependence and the partial linearity are investigated by comparing the behavior of our model to that of the nonspatial partially linear probit (NSPLP) model and the fully linear SAE probit (LSAEP) model, respectively. The GAM and ProbitSpatial [START_REF] Martinetti | Probitspatial: Probit with spatial dependence, sar and sem models[END_REF]) R packages will be used to provide the estimates associated to NSPLP and LSAEP models respectively. We generate observations from the following spatial latent partial linear model:

Y * in = β 1 X (1) in + β 2 X (2) in + g(Z in ) + U in ; Y in = I(Y * in > 0), i = 1, . . . , n U n = (I n -λW n ) -1 ε n
where U n ∼ N (0, I n ) and W n is the spatial weight matrix associated to n locations chosen randomly in a 60 × 60 regular grid and with elements constructed in such way that each location has at least 6 neighbors. The explanatory variables X (1) and X (2) are generated as pseudo B(0.7) and U [-2, 2], respectively, and the other explanatory variable Z is equal to the sum of 48 independent random variables, each uniformly distributed over [-0.25, 0.25]. Here, we use the non-linear function g(t) = t + 2 cos(0.5πt) and parameters

β 1 = -1, β 2 = 1.
Different spatial dependence parameters λ; 0.2 (weak spatial dependence) 0.5 and 0.8 (strong spatial dependence) are considered. Finally, the sample size effect is observed by considering n equals to 200, 400 and 800 with 300 replications of each simulation.

Our estimation procedure is applied with a Gaussian kernel K(t) = (2π -1/2 ) exp(-t 2 /2) and optimal bandwidth b n selected by [START_REF] Severini | Quasi-likelihood estimation in semiparametric models[END_REF]'s approach detailed previously.

We consider the trivial instrumental variables and two choices of matrix M n = I n which leads to the pseudo-maximum profile likelihood estimators (named PLSP 1) and a second choice M n given in (3.25) with components τ ij = w nij , the estimates obtained with this matrix choice are denoted PLSP 2. The second choice of the weight matrix allows to incorporate more information about the spatial dependance.

The results are given in Table 3.1, the columns titles Mean, Median and SD give the average, median and standard deviation, respectively, over these 300 replications associated with each estimation method.

In one hand, when we compare the estimators (PLSP 1 and PLSP2) based on our approach (PLSPM) with those based on the LSAEP model, we notice that the latter yields more biased estimators of the coefficients β 1 and β 2 . It makes sense that ignoring the partial linearity (see also Figure 3.1) weakens the quality of the estimation of the coefficients β 1 and β 2 .

On the other hand, note that the LSAEP and PLSP 1 estimates are similar in case of low spatial dependence (λ = 0.2) compare to large spatial dependence (λ = 0.8) framework. It makes sense that ignoring a high spatial dependence does not allow a model that does not account any spatial structure to find consistent estimates of the coefficients β 1 and β 2 and the smooth function g(•) (see Figure 3.1) . Note that the second choice of the weight matrix (estimates PLSP 2 ) allowed to improve the efficiency of the proposed estimates particularly in case of high spatial dependence (see PLSP 2 estimates in case of λ = 0.8). In contrast, it is less appropriate in case of low spatial dependance.

However, one may think of testing the intensity of the spatial dependence before applying the proposed model with a non identity weight matrix, using for instance Moran's test.

Discussion

In this manuscript, we have proposed a spatial semi-parametric probit model for identifying risk factors at onset and with spatial heterogeneity. The parameters involved in the models are estimated using weighted likelihood and generalised method of moment methods. A technique based on dependent random arrays facilitates the estimation and derivation of asymptotic properties, which otherwise would have been difficult to perform due to the complexity introduced by the spatial dependence to the model and high-dimensional integration required by a full maximum likelihood approach. Moreover, the technique yields consistent estimates through proper choices of the bandwidth, weight matrix, and instrumental variables. The proposed models provide a general framework and tools for researchers and practitioners when addressing binary semiparametric choice models in the presence of spatial correlation. Although they provide significant contributions to the body of knowledge, additional investigations need to be done.

As indicated previously, weights are used to improve the efficiency and convergence of the GMM procedure. For instance, the finite sample properties section shown that the kind of weight matrix defined in 3.25 with elements τ ij may improve the efficiency of the proposed estimator but is less appropriate in case of weak spatial dependence. Then, it would be interesting to develop other choices of weights τ ij toward achieving a better performance. Another topic of future research is to allow some spatial dependency in the covariates (SAR models) and the response (endogenous models) for more generality.

Table 3.1: The mean, median and standard deviation (SD) of the parameters β 1 , β 2 , and λ estimates, over the 300 replications 1. for all i, j = 0, 1, 2, i + j ≤ 2,

λ n Methods β 1 = -1 β 2 =
∂ i+j ∂θ i l ∂θ j r g θ (z) and ∂ i+j ∂θ i l ∂θ j r ĝθ (z)
exist and are finite for all 1 ≤ l, r ≤ p + 1.

sup

θ∈Θ ĝθ -g θ , sup θ∈Θ max j=1,...,p+1 ∂ ∂θ j (ĝ θ -g θ ) and sup θ∈Θ max 1≤i,j≤p+1 ∂ 2 ∂θ i ∂θ j (ĝ θ -g θ ) ,
are all order o p (1) as n → ∞.

Without loss of generality, the proof of this proposition is ensured by Lemma 3.5.2 in the univariate case i.e., Θ, Z ⊂ R.

The following lemma is useful in the proof of Lemma 3.5.2. It is an extension of Lemma 8 in [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF] to spatially dependent data.

Lemma 3.5.1. Let ζ θ (Y i ) denote a scalar function of Y in , i = 1, . . . , n, n = 1, 2, . . ., depending
on a scalar parameter θ ∈ Θ, and for j = 0, 1,

θ (Y in ) = ∂ j ∂θ j ζ θ (Y in ), i = 1, . . . , n, n = 1, 2, . . . Let f i (•) 2, let ζ (j)
denote the density of Z in (given in Assumption A2), and let

f (z) = 1 n n i=1 f i (z). Assume that H.1 sup θ sup 1≤i≤n,n ζ (j) θ (Y in ) < ∞ for j = 0, . . . , 3.
H.2 For all θ ∈ Θ, j = 0, 1, 2, and 1 ≤ i, l ≤ n:

|Cov (K in (z), K ln (z))| ≤ {Var(K in (z))Var(K in (z))} 1/2 ϕ ( s i -s l ) , (3.26) Cov ζ (j) θ (Y in )K in (z), ζ (j) θ (Y ln )K ln (z) ≤ Var ζ (j) θ (Y in )K in (z) Var ζ (j) θ (Y ln )K ln (z) 1/2 ϕ ( s i -s l ) , (3.27) with K in (z) = K ((z -Z in )/b). Let m θ (z) = E (ζ θ (Y in )|Z in = z) for z ∈ Z, and assume that ∂ j ∂θ j m θ (•) is continuous on Z, j = 0, 1, 2.
For each fixed θ ∈ Θ and z ∈ Z, let the kernel estimator m θ (z) of m θ (z) be defined by

m θ (z) = n i=1 ζ θ (Y in )K in (z) n i=1 K in (z) .
If Assumptions A2, A4, and A5 are satisfied, then

sup θ∈Θ sup z∈Z ∂ j ∂θ j m θ (z) - ∂ j ∂θ j m θ (z) = o p (1), for j = 0, 1, 2.
Lemma 3.5.1 generalizes Lemma 8 in [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF] to spatially dependent data.

Proof of Lemma 3.5.1

We give the proof in the case where j = 0, corresponding to the study of the uniform consistency of the kernel estimator of the regression function of ζ θ (Y in ) on Z in . The other cases are similar to this case and thus are omitted.

Let

v θ (z) = 1 nb d n i=1 ζ θ (Y in )K in (z); f (z) = 1 nb d n i=1 K in (z), v θ (z) = m θ (z) f (z).
We have to show that

sup θ sup z | v θ (z) -v θ (z)| = o p (1) (3.28)
and

sup z f (z) -f (z) = o p (1) (3.29)
We give the proof of (3.28), and that of (3.29) is similar.

Asymptotic behavior of | v θ (z) -v θ (z)| Let us first consider the bias |E( v θ (z)) -v θ (z)|.
We have

E( v θ (z)) = (nb d ) -1 n i=1 K z -u b m θ (u)f i (u)du = b -d v θ (u)K z -u b du; = v θ (z -bu)K(u)du thus, E( v θ (z)) -v θ (z) = (v θ (z -bu) -v θ (z)) K(u)du = o(1)
by Assumption A4, the continuity of f i (•) (see A2) and m θ (•), and the compactness of Z. Clearly, the bias term does not depend on θ or z.

Let us now treat

| v θ (z) -E( v θ (z))|.
Consider the sum of variances

S n = (nb d ) -2 n i=1 Var (ζ θ (Y in )K in (z)) .
3.5. Appendix Chapter 3. Partially linear spatial probit models

Using similar arguments as above, we have |B| ≤ Cb 2d by Assumptions A2 and A4, the compactness of Z and the continuity of m θ (•). Thus, we have

R (1) n ≤ Cn -2 i,j∈Dn ≤ C c 2 n -ρ 2 n = O c 2 n n . (3.32)
On the other hand, let

R (2) n = (n b d ) -2 i,j∈ Dn |Cov (ζ θ (Y in )K in (z), ζ θ (Y jn )K jn (z))| .
By Assumption H.2 combined with (3.30), we have for all θ ∈ Θ and i, j = 1, . . . , n,

|Cov (ζ θ (Y in )K in (z), ζ θ (Y jn )K jn (z))| ≤ C b d ϕ( s i -s j ).
Then, we have

R (2) n ≤ C(n b d ) -1 i>cn/ρ iϕ(iρ). (3.33)
Thus, we derive the following result:

R n = R (1) n + R (2) n = O   n -1    c 2 n + b -d i>cn/ρ iϕ(iρ)      . (3.34)
The following steps of the proof are inspired by the proof of Lemma 8 in [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF]) (p. 1800-1801). Let

ṽθ (z) = 1 n b -d n i=1 {ζ θ (Y in )K in (z) -E (ζ θ (Y in )K in (z))} .
For some > 0, Markov's inequality yields

P (|ṽ θ (z)| > ) ≤ R n + S n 2 . (3.35) Now, let θ 1 and θ 2 be two elements in Θ; because E sup θ,1≤i≤n,n |ζ (1) θ (Y in )| < ∞ (by H.1),
there exists a random triangular array (see [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF], p.1801)

W (1) in , 1 ≤ i ≤ n, n = 1, 2 . . . not depending on θ 1 and θ 2 such that sup 1≤i≤n, n E |W (1) in | < ∞ and sup z |ṽ θ 1 (z) -ṽθ 2 (z)| ≤ sup z |K(z)| |θ 2 -θ 1 | b d 1 n n i=1 W (1)
in .

Similarly, for all z (1) and z (2) in Z, there exists a random triangular array

W (2)
in , 1 ≤ i ≤ n, n = 1, 2 . . . not depending on z (1) and z (2) such that sup 1≤i≤n, n E |W

(2) i | < ∞ and sup θ ṽθ (z (2) ) -ṽθ (z (1) ) ≤ C z (2) -z (1) b d+1 1 n n i=1 W (2) in , because K(•) is Lipschitzian (see Assumption H.2).
Hence, there exists a random triangular array {W in ,

1 ≤ i ≤ n, n = 1, 2 . . .} such that sup 1≤i≤n, n E (|W in |) < 3.5. Appendix 51 ∞ and sup z (2) -z (1) <δ 1 sup |θ 2 -θ 1 |<δ 2 ṽθ 2 (z (2) ) -ṽθ 1 (z (1) ) ≤ C b -d δ 2 + b -(d+1) δ 1 1 n n i=1 W in ,
for some δ 1 > 0, δ 2 > 0 and large n.

Because Z is compact, one can define a real number δ 1 > 0, an integer l n such that l n δ 1 < C d+1) and

with l n = γ n b -(
Z ⊂ ln j=1 B(z (j) , δ 1 ),
where B(z, δ) is the closed ball in R d with center z and radius δ > 0.

In addition, because Θ is compact, one can cover it by r n = γ n b -d finite intervals of centers θ i with the same half length

δ 2 = O(1/r n ).
With these coverings, we have 3) , where

P sup θ,z |ṽ θ (z)| > ≤ P max j≤rn max k≤ln ṽθ j (z (k) ) > /2 + P sup z (2) -z (1) <δ 1 sup |θ 2 -θ 1 |<δ 2 ṽθ 2 (z (2) ) -ṽθ 1 (z (1) ) > /2 ≤ r n l n P (|ṽ θ (z)| > ε/2) + Cb -d δ 2 + δ 1 b -1 = C r n l n (S n + R n ) + Cb -d δ 2 + δ 1 b -1 := I (1) + I (2) + I (
I (1) = O   γ 2 n nb 2d+1   c 2 n + b -d i>cn/ρ iϕ(iρ)     ; I (2) = O γ -1 n ; I (3) = O γ 2 n nb 3d+1 . If we take c n = o(b -d/2
) and γ 2 n = o(nb 3d+1 ), then I (1) , I (2) and I (3) are all of order o(1) by Assumption A5 and by the fact that ϕ(t) → 0 as t → ∞ by Assumption A3. This yields the proof.

Lemma 3.5.2. For each θ ∈ Θ and z ∈ Z, let

H(η; θ, z) = E 0 h θ, η in (Y in |X in , Z in )|Z in = z , 1 ≤ i ≤ n, n = 1, 2, . . . where η = g(z), g ∈ G and h θ, η in (•|•, •) is defined in Assumption A3.
Condition I: For fixed but arbitrary θ 1 ∈ Θ and η 1 ∈ Π with Π = g 0 (Z), let

ϑ(θ, η) = h θ,η in (y|x , z) exp(h θ 1 ,η 1 in (y|x , z))dy, θ ∈ Θ, η ∈ Π, (x, z) ∈ Z × Z
where {exp(h θ,η in (y|x , z)), θ ∈ Θ, η ∈ Π} denotes the family of conditional density functions (indexed by the parameters θ and η) of

Y in given (X in , Z in ) = (x, z) ∈ X × Z. For each θ = θ 1 , assume that ϑ(θ, η) < ϑ(θ 1 , η 1 ).
Condition S: Let p = p+1, and for all nonnegative integers j 1 , . . . , j p = 0, 1, 2 and r = 0, . . . , 4, such that j 1 + • • • + j p + r ≤ 6, assume that the derivative

∂ j 1 +•••+j p+r h θ,η in ∂θ j 1 1 • • • ∂θ j p p ∂η r (y|x , z),
exists for almost all y and that

E 0   sup i, n sup θ∈Θ sup g∈G ∂ j 1 +•••+j p+r h θ,η i in ∂θ j 1 1 • • • ∂θ j p p ∂η r (Y in |X in , Z in ) 2   < ∞, with η i = g(Z in ).
Assume that

sup z sup θ sup η ∂ j ∂θ j H (k) (η; θ, z) < ∞, (3.36)
for j = 0, 1, 2 and k = 2, 3, 4 such that j + k ≤ 4, with

H (k) (η; θ, z) = ∂ k ∂η k H(η; θ, z). Let H(η; θ, z) = n i=1 h θ,η in (Y in |X in , z)K in (z) n i=1 K in (z)
; then, g θ (z) is a solution of H (1) (η; θ, z) = 0 with respect to η for each fixed θ ∈ Θ and z ∈ Z.

If we assume that Assumptions A1-A6 are satisfied, then we have, for all j = 0, 1, 2,

sup θ sup z ∂ j ∂θ j ( g θ (z) -g θ (z)) = o p (1). (3.37)
The assumptions used in the previous lemma are satisfied under the conditions used in the main results. Condition I is needed to ensure the identifiability of the arbitrary parameter θ 1 (it plays the role of the true parameter θ 0 ). This condition is verified when θ 1 = θ 0 by the identifiability of our model (3.1). Condition S allows integrals to be interchanged with differentiation; this will be combined with the implicit function theorem (see [START_REF] Saaty | Nonlinear mathematics[END_REF] to ensure the differentiability of ĝθ (z) with respect to θ.

Knowing that Φ(•) is a smooth function on R and h θ,η in (•|• , •) is h θ,η i in (Y in |X in , Z in ) = Y in log Φ(G in (θ, η i )) 1 -Φ(G in (θ, η i )) -log (1 -Φ(G in (θ, η i ))) ,
Condition S and Assumption (3.36) are satisfied under the continuity condition of Φ(•) and φ(•), Assumption A9 and the compactness of X and Z.

Appendix

53

Proof of Lemma 3.5.2

The proof of this lemma is similar to that of Lemma 5 in [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF]. Let us follow similar lines as in the proof of Lemma 3.5.1 above, replacing ζ

(j) θ (Y in ) by ζ (j,k) θ,η (Y in , X in ) = ∂ j ∂θ j ∂ k ∂η k h θ,η in (Y in |X in , z).
and Assumptions H.1 and H.2 in Lemma 3.5.1 by the following:

H.1' sup θ sup η sup i, n ζ (j,k) θ,η (Y in , X in ) < ∞, for j = 0, . . . , 3, k = 0, . . . , 5
H.2' For all k = 0, . . . , 4, j = 0, 1, 2 and θ ∈ Θ, z ∈ Z, (3.26) is satisfied and (3.27) holds with

ζ (j) θ (Y in ) replaced by ζ (j,k) θ,η (Y in , X in ).
Under the conditions used in the lemma, it is clear that H.1' is verified, and H.2' is also satisfied by Assumption A3 (in particular, conditions (3.19)).

Using the results of Lemma 3.5.1, we have the following for all j = 0, 1, 2:

sup θ, η, z ∂ j ∂θ j H (1) n (η; θ, z) -H (1) (η; θ, z) = o p (1), (3.38) sup θ, η, z ∂
j ∂θ j H (2) n (η; θ, z) -H (2) (η; θ, z) = o p (1), (3.39)
sup θ, η, z ∂ j ∂θ j H (3) n (η; θ, z) -H (3) (η; θ, z) = o p (1), (3.40) sup θ, η, z ∂ j ∂θ j H (4) n (η; θ, z) -H (4) (η; θ, z) = o p (1). (3.41)
Under Assumption A1, for any > 0, there exists γ > 0 such that

P sup θ,z | g θ (z) -g θ (z)| > ≤ P sup θ,z |H (1) (θ, g θ (z), z)| > γ = P sup θ,z | H (1) ( g θ (z); θ, z) -H (1) ( g θ (z); θ, z)| > γ ≤ P sup θ,z,η | H (1) (η; θ, z) -H (1) (η; θ, z)| > γ .
Hence,

sup θ,z | g θ (z) -g θ (z)| = o p (1) (3.42)
The remainder of the proof is very similar to that of Lemma 5 in [START_REF] Severini | Profile likelihood and conditionally parametric models[END_REF]) (p. 1798-1799); for the sake of completeness, we present the details.

We have by Condition I

inf θ inf z -H (2) (g θ (z); θ, z) > 0.
In addition, by Condition S, for every δ > 0, there exists > 0 such that

sup θ sup z sup η 1 ,η 2 :|η 1 -η 2 |≤ H (2) (η 2 ; θ, z) -H (2) (η 1 ; θ, z) < δ.

54

Chapter 3. Partially linear spatial probit models

Hence, there exists > 0 such that

inf θ inf z inf |η-g θ (z)|≤ H (2) (η; θ, z) > 0. (3.43)
Because g θ (z) and g θ (z) satisfy

H (1) (g θ (z); θ, z) = 0 and H (1) ( g θ (z); θ, z) = 0,
respectively, for each θ and z, it follows that

0 = H (1) ( g θ (z); θ, z) -H (1) (g θ (z); θ, z) = H (1) ( g θ (z); θ, z) -H (1) ( g θ (z); θ, z) + H (1) ( g θ (z); θ, z) -H (1) (g θ (z); θ, z) = r n (θ, z) + d n (θ, z) ( g θ (z) -g θ (z)) , (3.44)
for each θ, z, where

r n (θ, z) = H (1) ( g θ (z); θ, z)-H (1) ( g θ (z); θ, z) and d n (θ, z) = 1 0 H (2) (tg θ (z)+(1-t) g θ (z); θ, z)dt.
Note that by (3.43) and sup θ g θ -g θ = o p (1), we have

lim inf inf z inf θ H (2) ( g θ (z); θ, z) > 0 and lim inf inf z inf θ |d n (θ, z)| > 0 as n → ∞. (3.45)
Because

H (1) ( g θ (z); θ, z) = 0,
for all θ, z, we have

H (2) ( g θ (z); θ, z) ∂ g θ ∂θ (z) + ∂ H (1) ∂θ ( g θ (z); θ, z) = 0.
Then, we can deduce from (3.45), (3.38), and (3.39) that

sup θ sup z ∂ g θ ∂θ (z) = O p (1).
Similarly, we have (3.47) Now, differentiating (3.44) with respect to θ yields

sup θ sup z ∂ j g θ ∂θ j (z) = O p (1), j = 0, 1, 2. ( 3
∂r n ∂θ (θ, z) + ( g θ (z) -g θ (z)) ∂d n ∂θ (θ, z) + d n (θ, z) ∂ g θ ∂θ (z) - ∂g θ ∂θ (z) = 0. (3.48) Then, by (3.38)-(3.47), sup θ sup z ∂ g θ ∂θ (z) - ∂g θ ∂θ (z) = o p (1).

Appendix 55

On can similarly obtain

sup θ sup z ∂ 2 g θ ∂θ 2 (z) - ∂ 2 g θ ∂θ 2 (z) = o p (1).
This completes the proof.

Proof of Theorem 3.2.1

By Lemmas 3.5.3 and 3.5.4, Q n converges to Q in probability uniformly, i.e.,

sup θ∈Θ |Q n (θ, g θ ) -Q(θ, g θ )| = o p (1). (3.49)
This result allows one to obtain

Q( θ, g θ) -Q(θ 0 , g 0 ) = o p (1). (3.50) Indeed, using | sup a -sup b| ≤ sup |a -b|, we have Q( θ, g θ) -Q(θ 0 , g 0 ) ≤ Q n ( θ, ĝθ ) -Q( θ, g θ) + Q n ( θ, ĝθ ) -Q(θ 0 , g 0 ) ≤ sup θ |Q n (θ, ĝθ ) -Q(θ, g θ )| + sup θ Q n (θ, ĝθ ) -sup θ Q(θ, g θ ) ≤ 2 sup θ |Q n (θ, ĝθ ) -Q(θ, g θ )| ≤ 2 sup θ |Q n (θ, ĝθ ) -Q n (θ, g θ )| + 2 sup θ |Q n (θ, g θ ) -Q(θ, g θ )| = o p (1),
by Lemma 3.5.5,(3.49) and sup θ Q(θ, g θ ) = Q(θ 0 , g 0 ) (see Assumption A8).

By Assumption A8, we have for a given θ ∈ Θ that there exists ε > 0 and an open neighbourhood N θ such that inf (3.51) This and (3.50) imply that . By the compactness of Θ 0 , let {N θ 1 , . . . , N θr } be a finite sub-covering; then,

θ 1 ∈N θ |Q(θ 1 , g θ 1 ) -Q(θ 0 , g 0 )| > ε.
P 0 θ ∈ N θ ≤ P 0 Q( θ, g θ) -Q(θ 0 , g 0 ) > ε → 0, as n → ∞. ( 3
P 0 θ / ∈ N 0 = P 0 θ ∈ Θ 0 ≤ r j=1 P 0 θ ∈ N θ j → 0, as n → ∞,
by (3.52). Therefore, we can conclude that

θ -θ 0 = o p (1), as n → ∞.
This yields the proof of Theorem 3.2.1. We use the following notation:

η i = g(Z in ); Ũin = Ũin (θ, η i ); Φ in = Φ(G in (θ, g θ )); Λ in = Λ(G in (θ, g θ )), for all θ ∈ Θ, 1 ≤ i ≤ n, n = 1, 2, . . ., with Λ(•) = φ(•)/Φ(•)(1 -Φ(•)).
The partial derivatives of S n (θ, g) with respect to g of order s = 1, 2, . . ., for any functions v 1 , . . . , v s in G, are given by

∂ s S n ∂g s (θ, g)(v 1 , • • • , v s ) = n -1 n i=1 ξ in ∂ s Ũin ∂η s (θ, η i )v 1 (Z in ) • • • v s (Z in ).
Lemma 3.5.3. Under Assumptions A3, A6 and A9, we have for all θ ∈ Θ,

S n (θ, g θ ) -S (θ, g θ ) = o p (1). (3.53)
In addition, we have

Q n (θ, g θ ) -Q (θ, g θ ) = o p (1), (3.54) if M n -M = o p (1). Note that if Assumption A10 is satisfied, then M n -M = o p (1).
Proof of Lemma 3.5.3

Let us start with the proof of (3.53). We remark that

S n (θ, g θ ) = n -1 ξ T n Ũn (θ, g θ ) = n -1 n i=1 ξ in Ũin (θ, g θ ),
where ξ i is the q × 1 vector representing the ith row in the matrix of instrumental variables. By definition (see (3.13)), we have E 0 (S n (θ, g θ )) -S(θ, g θ ) = o(1). Then, it suffices to show that

S n (θ, g θ ) -E 0 (S n (θ, g θ )) = o p (1). (3.55)
Indeed (omitting the (θ, g θ )-arguments to simplify the notation), we have

E 0 S n -E 0 (S n ) 2 = n -2 n i,j=1 E 0 ξ in Ũin -E 0 (ξ in Ũin ) T ξ jn Ũjn -E 0 (ξ jn Ũjn ) (3.20) ≤ n -2 n i,j=1 α ijn q t=1
Var 0 ξ itn Ũin Var 0 ξ jtn Ũjn

1/2 ≤ Cn -2 n i,j=1 α ijn = O   n -1 √ n s=1 sϕ(s)   = o(1),
because Var 0 (ξ itn Ũin ) is bounded uniformly on θ, i, and t = 1, . . . , q (by Assumption A6) and because ϕ(s) → as s → +∞ (by assumption A3). This completes the proof of (3.55) and thus 3.5. Appendix 57 that of (3.53).

The proof of (3.54) is made straightforward by combining (3.53) with Assumption A10.

Lemma 3.5.4. Under Assumptions A6-A9, we have S n (•, g • ) -S (•, g • ) is stochastically equicontinuous on Θ.

In addition, if M n -M = o p (1), then we have

Q n (•, g • ) -Q (•, g • ) is also stochastically equicon- tinuous on Θ.
Proof of Lemma 3.5.4 Stochastic equicontinuity in Θ can be obtained by proving that S n (θ, g θ ) satisfies a stochastic Lipschitz-type condition on θ (see Mátyás, 1999, p. 17).

Let us show that S n (•, g • ) is stochastically equicontinuous on θ because S(•, g • ) is continuous by Assumption A8. It suffices to show that [START_REF] Andrews | Generic uniform convergence[END_REF] for each θ 1 , θ 2 ∈ Θ:

S n (θ 1 , g θ 1 ) -S n (θ 2 , g θ 2 ) = O p ( θ 1 -θ 2 ) . (3.56) Indeed, for θ 1 , θ 2 ∈ Θ, S n (θ 1 , g θ 1 ) -S n (θ 2 , g θ 2 ) ≤ n -1 sup i, n ξ in n i=1 Ũin (θ 1 , g θ 1 ) -Ũin (θ 2 , g θ 2 ) ≤ n -1 sup i, n ξ in n i=1 sup θ, η ∂ Ũin ∂θ (θ, η) θ 1 -θ 2 + sup θ, η ∂ Ũin ∂η (θ, η) g θ 1 -g θ 2 ≤ n -1 sup i, n ξ in n i=1 sup θ, η ∂ Ũin ∂θ (θ, η) + sup θ ∂g θ ∂θ sup θ, η ∂ Ũin ∂η (θ, η) θ 1 -θ 2 .
By Assumption A6 and Proposition 3.5.1, we have that sup i, n ξ in is bounded and sup θ ∂g θ ∂θ is finite, respectively. Then, we have to show that Let us prove (3.58) in the following. The proof of (3.59) follows the same lines and is thus omitted.

n -1 n i=1 sup θ,η ∂ Ũin ∂θ (θ, η) + sup θ,η ∂ Ũin ∂η (θ, η) = O p (1); (3.57) This is equivalent to sup θ,η ∂ Ũin ∂θ (θ, η) = O p (1), 1 ≤ i ≤ n, n = 1, 2, . . . ( 3
Proof of (3.58):

Recall that

Λ(t) = φ(t) Φ(t)(1 -Φ(t))
.

By definition, we have

Ũin (θ, η) = Λ(G in (θ, η)) (Y in -Φ(G in (θ, η))) ,
with G in (θ, η) = a in (θ)b in (θ, η), where a in (•) and b in (•) are defined by

a in (θ) := (v in (λ)) -1 and b in (θ, η) := X T in β + η, 1 ≤ i ≤ n, n = 1, 2, . . . , (3.60)
with θ T = (β T , λ). We have

∂ Ũin ∂θ (θ, η) = Λ (G in (θ, η))(Y in -Φ(G in (θ, η))) -Λ(G in (θ, η))φ(G in (θ, η))} ∂G in ∂θ (θ, η) (3.61)
where Λ (•) denotes the derivative of Λ(•).

Let us first establish that sup t∈M,y∈{0,1}

Λ (t)(y -Φ(t)) -φ(t)Λ(t) < ∞, (3.62)
which is equivalent to showing that Λ (t) and φ(t)Λ(t) are bounded uniformly in t ∈ M (the definition of M is given in A.1). Because φ (t) = -tφ(t), we can rewrite Λ (t) as .63) Notice that Λ(•) and Λ (•) may be unbounded only at ±∞, and because M is a compact subset of R, these functions are bounded on R. This establishes (3.62).

Λ (t) = 1 Φ(t) φ(t) 1 -Φ(t) φ(t) 1 -Φ(t) -t - φ 2 (t) Φ 2 (t)(1 -Φ(t)) . ( 3
We remark that

∂G in (θ, η) ∂θ ≤ ∂a in (θ) ∂θ |b in (θ, η)| + ∂b in (θ, η) ∂θ |a in (θ)| . ( 3

Proof of Theorem 3.2.2

Recall that d dθ Q n (θ, g θ ) denotes differentiation with respect to θ, while ∂ ∂θ Q n (θ, g θ ) denotes the partial derivative with respect to θ.

Using a Taylor's series expansion and the fact that

d dθ Q n (θ, ĝθ ) θ= θ = 0,
we have

θ -θ 0 = - d 2 dθdθ T Q n (θ, ĝθ ) θ=θ * -1 d dθ Q n (θ, ĝθ ) θ=θ 0 , (3.67)
for some θ * between θ 0 and θ.

First, we would like to replace ĝθ (.) in (3.67) with g θ (.). For this, let us show that d dθ Q n (θ, ĝθ ) (resp.

d 2 dθdθ T Q n (θ, ĝθ )) and d dθ Q n (θ, g θ ) (resp. d 2 dθdθ T Q n (θ, g θ ))
have the same behavior as a function of θ in a neighbour of θ 0 . In other words, (see Lemma 3.5.5).

sup θ d 2 dθdθ T Q n (θ, ĝθ ) - d 2 dθdθ T Q n (θ, g θ ) = o p (1) (3.68) and d dθ Q n (θ, ĝθ ) θ=θ 0 - d dθ Q n (θ, g θ ) θ=θ 0 = o p (1). ( 3
Then, (3.70) and (3.71) follow immediately from Lemma 3.5.8.

To prove (3.69), we have the following Taylor expansion

d dθ (Q n (θ, ĝθ ) -Q n (θ, g θ )) = d dθ ∂Q n ∂g (θ, g θ )(ĝ θ -g θ ) + rn (θ) ,
where

rn (θ) = 1 0 ∂ 2 Q n ∂g 2 (θ, g θ + t(ĝ θ -g θ ))(ĝ θ -g θ ) 2 dt.
We have d dθ rn (θ)

θ=θ 0 = o p (1),
using similar arguments as for the terms d j dθ j r (1) n (θ) for j = 0, 1 and

d 2 dθdθ T r (1)
n (θ) in Lemma 3.5.8 below (see (3.89)). Therefore, we obtain

d dθ Q n (θ, ĝθ ) θ=θ 0 - d dθ Q n (θ, g θ ) θ=θ 0 = d dθ ∂Q n ∂g (θ, g θ ) θ=θ 0 (ĝ 0 -g 0 ) + ∂Q n ∂g (θ 0 , g 0 )(ĝ 0 -g 0 ) + d dθ r n (θ) θ=θ 0 , = o p (1)
by Lemma 3.5.7, where g 0 (.) = g θ ∂θ T (.) θ=θ 0

.

Consequently, we obtain

θ -θ 0 = - d 2 dθdθ T Q n (θ, g θ ) θ=θ * -1 d dθ Q n (θ, g θ ) θ=θ 0 + o p (1) (3.72)
where θ * is between θ and θ 0 .

Let us show that for each θ * lying between θ 0 and θ,

d 2 dθdθ T Q n (θ, g θ ) θ=θ * = 2 B 2 (θ 0 ) + o p (1),
to replace the Hessian matrix in the right-hand side of (3.72) by its limit B 2 (θ 0 ).

Let us consider the first-and second-order differentials of Q n (θ, g θ ) with respect to θ:

d dθ Q n (θ, g θ ) = 2S T n (θ, g θ )M n ∂S n ∂θ (θ, g θ ) + ∂S n ∂g (θ, g θ )g θ (3.73)
with g θ being a 1 × p (p = p + 1) matrix given by ∂g θ ∂θ T and

d 2 dθdθ T Q n (θ, g θ ) = 2 ∂S n ∂θ (θ, g θ ) + ∂S n ∂g (θ, g θ )g θ T M n ∂S n ∂θ (θ, g θ ) + ∂S n ∂g (θ, g θ )g θ +2S T n (θ, g θ )M n d dθ T ∂S n ∂θ (θ, g θ ) + ∂S n ∂g (θ, g θ )g θ (3.74) 3.5. Appendix 61 with d dθ T ∂S n ∂θ (θ, g θ ) = ∂ 2 S n ∂θ∂θ T (θ, g θ ) + ∂ 2 S n ∂θ∂g (θ, g θ )g θ , d dθ T ∂S n ∂g (θ, g θ ) = ∂ 2 S n ∂θ∂g (θ, g θ ) + ∂ 2 S n ∂g 2 (θ, g θ ) ∂g θ ∂θ .

Note that

S n (θ * , g θ * ) = S n (θ * , g θ * ) -S n (θ 0 , g 0 ) + S n (θ 0 , g 0 ) -S(θ 0 , g 0 ) = o p (1), because S(θ 0 , g 0 ) = 0 and by ,

S n (θ 0 , g 0 ) -S(θ 0 , g 0 ) = o p (1),
and because θ * lies between θ and θ 0 , by Lemma 3.5.4

S n (θ * , g θ * ) -S n (θ 0 , g 0 ) = o p (1).

Using similar arguments as in the proof of (3.58) in Lemma 3.5.4 using Assumption A9 to ensure the boundedness when differentiating twice with respect to θ, we have

d dθ T ∂S n ∂θ (θ, g θ ) = O p (1) and d dθ T ∂S n ∂g (θ, g θ )g θ = O p (1). (3.75)
Then, we can ignore the second term in the right-hand side of (3.74) at θ = θ * . Hence, by Lemma 3.5.6 and θ * -θ 0 = o p (1) (thanks to Theorem 3.2.1), we have

∂S n ∂θ (θ * , g θ * ) - ∂S ∂θ (θ 0 , g 0 ) = o p (1)
and

∂S n ∂g (θ * , g θ * )g θ * - ∂S ∂g (θ 0 , g 0 )g 0 = o p (1), with g θ * = g θ ∂θ T θ=θ * .
In addition, if M n -M = o p (1), we deduce that

d 2 dθdθ T Q n (θ, g θ ) θ=θ * = 2 ∂S ∂θ (θ 0 , g 0 ) + ∂S ∂g (θ 0 , g 0 )g 0 T M ∂S ∂θ (θ 0 , g 0 ) + ∂S ∂g (θ 0 , g 0 )g 0 + o p (1) = 2 B 2 (θ 0 ) + o p (1).
We remark that

d dθ Q n (θ, g θ ) θ=θ 0 = 2S T n (θ 0 , g 0 )M n ∂S n ∂θ (θ 0 , g 0 ) + ∂S n ∂g (θ 0 , g 0 )g 0 .
Then, by (3.79) (see the proof of Lemma 3.5.6), we have Assumption C. For some τ > 1

∂S n ∂θ (θ 0 , g 0 ) - ∂S ∂θ (θ 0 , g 0 ) = o p (
E 0 |A jtn | 2τ = o σ 2 jtn γ 2τ -2
jn , j = 1, . . . , J, t = 1, . . . , m jn .

If assumptions A -C hold, then by Theorem 1 in [START_REF] Pinkse | A central limit theorem for endogenous locations and complex spatial interactions[END_REF], we have A n -→ N (0, 1).

Thus, to complete the proof, we have to check these assumptions in our context.

Assumption A: This holds under (3.20) (Assumption A3).

Let us choose for instance J = 2 groups, each with m 1n , m 2n subgroups such that m 2n = o(m 1n ).

Each subgroup is viewed as an area of size O(

√ c n × √ c n ) such that (m 1n + m 2n )c n = O(n). Because ϕ(•) is a decreasing function (Assumption A3), α jn = O(ϕ( √ c n )) for j = 1, 2.
The sequence c n must be such that c n = O(n -ν+1/2 ) for some 0 < ν < 1/2 and n ν+1/2 ϕ(

√ c n ) → 0 as n → ∞. If for instance ϕ(t) = O(t -ι ), then n ν+1/2 ϕ( √ c n ) = O(n ι(ν-1/4)+(1+ν)/2
); this tends to 0 for each

ι > 2(1 + ν)/(1 -4ν).
Assumption B : By assumption A10, B 1 (θ 0 ) is positive definite and by definition is the limit of E 0 nS n (θ 0 , g 0 )S T n (θ 0 , g 0 ) . Then, for sufficiently large n, the last matrix is positive definite, and its inverse is O(1). Therefore, B in is bounded uniformly on i and n because ξ in is bounded uniformly on i and n by Assumption A6, as is Ũin (θ 0 , g 0 ). Then, for all j = 1, . . . , J and t = 1, . . . , m nj ,

σ jtn =    n -1 E 0   i∈G jtn B in      1/2 = O n -1/2 Card(G jtn )
and

γ jn = O m jn √ n max t≤m jn Card(G jtn ) .
Therefore,

σ jtn γ jn = O(1/m jn ) → 0 as n → ∞,
for all j = 1, . . . , J and t = 1, . . . , m jn . Now, consider the second limit in Assumption B. We have for all j = 2, . . . , J

γ jn γ 1n = O m jn max t≤m jn Card(G jtn ) m 1n max t≤m 1n Card(G 1tn ) = O m jn m 1n → 0 as n → ∞,
because m jn /m 1n = o(1) for all j = 2, . . . , J as n → ∞.

Assumption C : By an easy calculation, we can show that

E 0 |A jtn | 2τ σ 2 jtn γ 2τ -2 jn = O(m 2-2τ jn ) → 0 as n → ∞.
Lemma 3.5.6. Under the assumptions of Theorem 3.2.2 and for any θ such that θ -θ 0 = o p (1),

we have ∂S n ∂θ ( θ, g θ) - ∂S ∂θ (θ 0 , g 0 ) = o p (1) (3.76) and ∂S n ∂g ( θ, g θ)g θ - ∂S ∂g (θ 0 , g 0 )g 0 = o p (1), (3.77)
with g θ(.) = g θ ∂θ T (.) θ= θ.

Proof of Lemma 3.5.6 To prove (3.76), we need to show that for all w ∈ R q with w = 1,

w T ∂S n ∂θ ( θ, g θ) - ∂S ∂θ (θ 0 , g 0 ) = o p (1)
, which is equivalent to

w T ∂S n ∂θ ( θ, g θ) - ∂S n ∂θ (θ 0 , g 0 ) = o p (1) (3.78)
and

w T ∂S n ∂θ (θ 0 , g 0 ) - ∂S ∂θ (θ 0 , g 0 ) = o p (1). (3.79)
The proof of (3.78) is similar to that of (3.56), using the fact that Now, let us prove (3.79). By the definition of S(• , •) (see 3.13)

lim n→∞ E 0 ∂S n ∂θ (θ 0 , g 0 ) = ∂S ∂θ (θ 0 , g 0 ).
Thus, it suffices to prove that

w T ∂S n ∂θ (θ 0 , g 0 ) -w T E 0 ∂S n ∂θ (θ 0 , g 0 ) = o p (1). (3.80) 3.5. Appendix 65 Let w T ∂S n ∂θ (θ 0 , g 0 ) = n -1 w T ξ in ∂ Ũin ∂θ (θ 0 , η 0 i ), = ∆ n1 -∆ n2 , (3.81)
where

∆ n1 = n -1 n i=1 ξ (1) in (θ 0 , η 0 i ) Y in -Φ G in (θ 0 , η 0 i ) and ∆ n2 = n -1 n i=1 ξ (2) in (θ 0 , η 0 i ), with ξ (1) in (θ 0 , η 0 i ) := w T ξ i Λ G in (θ 0 , η 0 i ) ∂G i ∂θ (θ 0 , η 0 i ), ξ (2)
in (θ 0 , η 0

i ) := w T ξ in Λ G in (θ 0 , η 0 i ) φ G in (θ 0 , η 0 i ) ∂G in ∂θ (θ 0 , η 0 i ),
and η 0 i = g 0 (Z in ). The proof of (3.80) is then reduced to proving

E 0 ∆ n1 2 = o(1) and E 0 ∆ n2 -E 0 (∆ n2 ) 2 = o(1). (3.82)
This last part is trivial because ξ

(1)

in and ξ

( 2) in are bounded uniformly on i and n (see Assumption A6 and the compactness of Θ, X , and Z) and by use of the mixing condition (3.20) and (3.21) in Assumption A3. This completes the proof of (3.76).

To prove (3.77), we remark that

∂S n ∂g ( θ, g θ)g θ - ∂S ∂g (θ 0 , g 0 )g 0 = ∂S n ∂g ( θ, g θ) - ∂S ∂g (θ 0 , g 0 ) g θ + ∂S ∂g (θ 0 , g 0 ) g θ -g 0 . ( 3
= o p (1), ∂S ∂g (θ 0 , g 0 ) g θ -g 0 = ( θ -θ 0 ) O ∂S ∂g (θ 0 , g 0 ) sup θ sup z ∂g θ (z) ∂θ∂θ T = o p (1).
For the first term on the right-hand side in (3.83), because g θ = O p (1) by Proposition 3.5.1, using similar arguments as when proving (3.76) permits one to obtain

∂S n ∂g ( θ, g θ) - ∂S ∂g (θ 0 , g 0 ) = o p (1).
This yields the proof of (3.77).

Lemma 3.5.7. Under the assumptions of Theorem 3.2.2, we have Proof of Lemma 3.5.7

(i) d dθ ∂Q n ∂g (θ, g θ ) θ=θ 0 (ĝ 0 -g 0 ) = o p (1) (ii) ∂Q n ∂g (θ, g θ ) θ=θ 0 (ĝ 0 -g 0 ) = o p (
To prove (i), and we note that

d dθ ∂Q n ∂g (θ, g θ ) = 2 d dθ S T n (θ, g θ )M n ∂S n ∂g (θ, g θ ) = 2 d dθ S T n (θ, g θ )M n ∂S n ∂g (θ, g θ ) + 2S T n (θ, g θ )M n d dθ ∂S n ∂g (θ, g θ ).
One can easily see that

d dθ S n (θ, g θ ) = ∂S n ∂θ (θ, g θ ) + ∂S n ∂g (θ, g θ )g θ and d dθ ∂S n ∂g (θ, g θ ) = ∂ 2 S n ∂θ∂g (θ, g θ ) + ∂ 2 S n ∂g 2 (θ, g θ )g θ .
Therefore, we have

d dθ ∂Q n ∂g (θ, g θ ) θ=θ 0 (ĝ 0 -g 0 ) = 2S T n (θ 0 , g 0 )M n ∂ 2 S n ∂θ∂g (θ 0 , g 0 ) + ∂ 2 S n ∂g 2 (θ 0 , g 0 )g 0 (ĝ 0 -g 0 ) + 2 ∂S n ∂g (θ 0 , g 0 )M n ∂S n ∂θ (θ 0 , g 0 ) + ∂S n ∂g (θ 0 , g 0 )g θ (ĝ 0 -g 0 ).
By Lemma (3.5.3) and S(θ 0 , g 0 ) = 0, we obtain S n (θ 0 , g 0 ) = S n (θ 0 , g 0 ) -S(θ 0 , g 0 ) = o p (1).

(3.84)

In addition, we have

∂ 2 S n ∂θ∂g (θ 0 , g 0 )(ĝ 0 -g 0 ) = n -1 ξ in ∂ 2 Ũin ∂θ∂η (θ 0 , η i )(ĝ 0 (Z in ) -g 0 (Z in )) ≤ n -1 sup i,n ξ in sup η ∂ 2 Ũin ∂θ∂η (θ 0 , η) ĝ0 -g 0 = o p (1), (3.85)
because ξ i is bounded uniformly on i, n and θ (Assumption A6), ĝ0 -g 0 = o p (1) by Proposition 3.5.1, and

sup i, n sup η ∂ 2 U in ∂θ∂η (θ 0 , η) < ∞.
Using similar arguments as in the proof of (3.85), we obtain

∂ 2 S n ∂g 2 (θ 0 , g 0 )(ĝ 0 -g 0 )g 0 = n -1 ξ i ∂ 2 U in ∂η 2 (θ 0 , η i )(ĝ 0 (Z in ) -g 0 (Z in ))g 0 (Z in ) = o p (1), (3.86) 3.5. Appendix 67 ∂S n ∂g (θ 0 , g 0 )(ĝ 0 -g 0 )g 0 = n -1 ξ in ∂U in ∂η (θ 0 , η i )(ĝ 0 (Z in ) -g 0 (Z in ))g 0 (Z in ) = o p (1), (3.87)
and

∂S n ∂θ (θ 0 , g 0 )(ĝ 0 -g 0 ) = n -1 ξ in ∂U in ∂θ (θ 0 , η i )(ĝ 0 (Z in ) -g 0 (Z in )) = o p (1). ( 3
(ĝ 0 -g 0 ) = o p (1).
This yields the proof of (i).

The proof of (ii) follows along similar lines as (i) and hence is omitted.

Lemma 3.5.8. Under the assumptions of Theorem 3.2.2, we have

S n (θ, ĝθ ) -S n (θ, g θ ) = r (1) n (θ),
where

sup θ ∂ ∂θ r (1) n (θ) = o p (1), and sup θ ∂ 2 ∂θ∂θ T r (1) n (θ) = o p (1)
Proof of Lemma 3.5.8

By applying Taylor's theorem to Ũi (θ, •) for each θ ∈ Θ, we obtain

S n (θ, ĝθ ) -S n (θ, g θ ) = n -1 n i=1 ξ in Ũin (θ, ĝθ ) -Ũin (θ, g θ ) = n -1 n i=1 ξ in (ĝ θ (Z in ) -g θ (Z in )) × 1 0 ∂ Ũin ∂η (θ, g θ (Z in ) + t (ĝ θ (Z in ) -g θ (Z in ))) dt := r (1) n (θ).
Because the instrumental variables are bounded uniformly on i, n, and θ (Assumption A6), Chapter 4

Application of spatial models to investigate suicide recidivism in Nord-Pas-de-Calais.

Introduction

According to the World Health Organization (WHO), more than 800000 people die from suicide every year, including 10000 in France, preventing suicide is a major public health issue. In 2013, the World Health Assembly adopted the first-ever Mental Health Action Plan of the World Health Organization. This plan aimed to reduce the rates of suicide by 10% from 2013 to 2020.

The understanding of risk factors involved in suicidal behavior is crucial for the development of effective prevention plans. Interestingly, one of the most robust risk factors for death by suicide is a history of previous suicide attempts, as a substantial number of patients who attempt a suicide ultimately die by suicide.

Suicide is a serious global public health issue and is among the leading causes of death worldwide. In 2019, more than one in every 100 deaths (1.3%) were the result of suicide. A previous suicide attempt is one of the most important contributory factors of future suicide (Hawton and Heeringen, 2009) as well as death by suicide [START_REF] Brent | Suicidal behavior runs in families: A controlled family study of adolescent suicide victims[END_REF]. [START_REF] Hulten | Repetition of attempted Bibliography suicide among teenagers in europe: Frequency, timing and risk factors[END_REF] investigated the repetition of attempted suicide among young people aged 15-19 years in some European countries and identified relevant factors associated with repeated suicidal behaviour. Many studies have proven that postcard or telephone follow-up helped to prevent reattempts [START_REF] Beautrais | Postcard intervention for repeat self-harm: Randomised controlled trial[END_REF][START_REF] Carter | Postcards from the edge project: Randomised controlled trial of an intervention using postcards to reduce repetition of hospital treated deliberate self poisoning[END_REF][START_REF] Cedereke | Telephone contact with patients in the year after a suicide attempt: Does it affect treatment attendance and outcome? a randomised controlled study[END_REF][START_REF] Evans | Crisis card following self-harm: 12-month follow-up of a randomised controlled trial[END_REF][START_REF] Exbrayat | Effect of telephone follow-up on repeated suicide attempt in patients discharged from an emergency psychiatry department: A controlled study[END_REF][START_REF] Guillaume | Effect of telephone contact on further suicide attempts in patients discharged from an emergency department: Randomized controlled study[END_REF][START_REF] Motto | A randomized controlled trial of postcrisis suicide prevention[END_REF][START_REF] Vaiva | Algos: The development of a randomized controlled trial testing a case management algorithm designed to reduce suicide risk among suicide attempters[END_REF]. [START_REF] Exbrayat | Effect of telephone follow-up on repeated suicide attempt in patients discharged from an emergency psychiatry department: A controlled study[END_REF] verified that a protocol of early telephone follow-up after attempted suicide is helpful in preventing reattempts.

In France, the VigilanS healthcare system is an effort to support those who have attempted suicide in various regions. It was established in the Nord-Pas-de-Calais region in February 2015.

This programme to monitor and prevent recidivism of suicide attempts is executed via phone calls by teams of professionals who are specialized in this type of remote care. At Lille University Hospital, this six-month programme is managed by the adult psychiatry department under Professor Guillaume Vaiva. Posthospital support is offered to those patients who attempted suicide.

A patient discharged from the hospital is given a resource card which includes the single regional call number. The patient is also given an information note indicating the terms and recidivism in Nord-Pas-de-Calais.

conditions of the system as well as his or her right to object. At the same time, the partner center notifies the VigilanS secretariat of the patient's discharge and entry into the surveillance system by sending an encrypted fax or email, or via a Web interface. The VigilanS secretariat opens a follow-up file and sends an information letter to the care partners with the contact number for health professionals (dedicated line).

• Non-primary suicidal patients are called by telephone between days 10 and 21 after their discharge from the hospital by the follow-up team.

• For all suicidal persons benefiting from the system, a telephone contact is scheduled at the end of the sixth month following discharge from hospital with a view to a clinical assessment and evaluation with a proposal for ending the monitoring. If necessary, the monitoring can be extended. For this call, the patient is informed 1 week before by a letter.

• In VigilanS, the suicidal person leaves the hospital with a resource card containing the toll-free number of the contact unit, which he or she can call at any time during opening hours. Many incoming calls are received; these are generally long calls from patients in need of help.

• The sending of personalized postcards; once a month for 4 months, can be decided by the contactor after any phone call and for unreachable patients.

The aim of this chapter is to:

1. identify the recidivism risk factors in the Nord-Pas-de-Calais region;

2. model the suicide recidivism by applying several frameworks of the probit regression model;

3. investigate the impact of non-linear explanatory variables (exogenous determinants) such as median revenue, unemployment rates, worker rates and graduation (high school graduates) rate.

The spatial autocorrelation factor as well as the non-linear explanatory variables are then incorporated to the partially linear spatial probit model [START_REF] Bibliography Ahmed | Partially Linear Spatial Probit Models[END_REF] to improve the accuracy of predicting suicide recidivism in Nord-pas-de-Calais. This model can aid in developing strategies to combat suicide death in a population.

Data description

The data consists of information on 34000 cases of suicide attempts from January 2015 to May 2021 for over 20000 patients. Among the suicide attempts, 31.3% entries show reattempts (22.95% of the patients). The reattempt cases will be called as feature recidivism and treated as our target variable. REC denotes recidivism at the point of recruitment. A target variable called Recidivism after 6 months denoted by REC6 was created for this data. REC6 is a binary variable which tells if the patient attempted suicide after the 6-month-call. It was created using the following logic:

• for a patient with single entry: REC = No,

• for a patient with multiple entries: REC = Yes for all but the last entry No.

Then, based on the date of call made at the 6th month, the duration between two successive acts was calculated and classified as follows for the post 6-month study:

Methods

Modeling the suicide attempt by taking into account potential spatial dependency requires to find a correlation structure between data observed at a given location and that available at neighboring locations. In spatial econometrics, spatial dependency is usually modeled by using a spatial linear process defined by a spatial weight matrix which is a n × n non-stochastic weight matrix, W n describing the spatial interactions between n spatial units. For a lattice data, the spatial autoregressive (SAR) dependent variable model and the spatial autoregressive error model (SAE).

Our goal is to model suicide recidivism for patients enrolled in the VigilanS programme. The response variable is a binary variable so we use the probit regression model. The integration space-dependent correlation to the model (if exists) further enhances the ability of the model by increasing its accuracy. Space-dependent correlation is detected when data observed between neighbouring locations form a correlation structure. The inclusion of the spatial index based on the location of the address reported for each patient lead to the use of the spatial probit model.

Suppose we have a sample of n observations collected from points in a region of interest located on an irregularly spaced, countable lattice I ⊂ R N , N ≥ 2. Let (Y s i , X s i ) i=1,...,n be a sequence of spatially dependent observations at these spatial n points denoted s i ∈ R N drawn from lattice I. Assume that all sites in I are located at distances of at least d > 0 for each other; i.e ∀ s i , s j ∈ I: s i -s j ≥ d. In this section, to facilitate the notation, we denote i for individual in location s i . The variables Y i are binary responses (Y i = 1 correspond to recidivism while 0 is no recidivism). Let X n be a n × p matrix of p exogenous discrete or continuous random variables with elements X ij , i = 1, . . . , n, j = 1, . . . , p. Suppose two alternatives for each observation is based on a latent dependent variable Y * i via the following spatial autoregressive regression:

Y * n = λ 0 W n Y * n + X n β 0 + ε n , ε n ∼ N (0, I n ), Y i = I (Y * i ≥ 0) , i = 1, . . . , n. (4.1)
Chapter 4. Application of spatial models to investigate suicide recidivism in Nord-Pas-de-Calais.

where the coefficient λ 0 is a scalar autoregressive parameter indicating the degree of spatial dependence, β 0 is a p × 1 vector of parameters. W n is a spatial weight matrix described by one of previous methods given in Chapter 1. Assume that the n × n matrix (I n -λ 0 W n ) is nonsingular for all n, therefore the variance-covariance matrix of the latent dependent vector of variables

Y * n is V n (λ 0 ) = Var (Y * n |X n ) = (I n -λ 0 W n ) -1 (I n -λ 0 W n ) -1
.

The ProbitSpatial R package was used to provide the estimates of β 0 and λ 0 in the following where the results are based on the spatial weight matrix W n with K-nearest neighbour (KNN)

weights.

The SAR probit model will be compared with the classical binary probit model (does not consider spatial dependence). We will present the numerical results for the spatial probit SAR (model 4.1)

and the following basic non-spatial binary probit model:

Y * n = X n β 0 + ε n , ε n ∼ N (0, I n ), Y i = I (Y * i ≥ 0) , i = 1, . . . , n. (4.2)

Partially linear spatial probit model

We consider that at n spatial locations {s 1 , s 2 , . . . , s n } satisfying s i -s j > ρ with ρ > 0, observations of a random vector (Y, X, Z) are available. Assume that these observations are considered as triangular arrays [START_REF] Robinson | Asymptotic theory for nonparametric regression with spatial data[END_REF]

Results and discussion

Bivariate analysis (Tables 4.8,4.9,4.10,4.11) showed statistically significant differences in the potential predictors of suicide recidivism. P-value less than 0.05 indicates statistical significant difference in the category studied.

Females consist about 61% of the total patients enrolled with those in the adolescent phase as well as in their 20s contributing the highest number of cases. On the other hand, for males, patients in their 40s and 50s contribute to the highest number of suicide cases.

This study also shows that alcohol consumption prior to a suicide attempt contributes to suicide recidivism (more than 50% of the patients, Table 4.2). Patients aged 40 and above have higher tendency in consuming alcohol before the act of committing suicide (Figure 4.5). Alcohol consumption among males (60%) is higher than that of females (40%) based on findings.

Patients use various methods for attempting suicide where VDI was used by almost 80% of females and 70% of males. Males tend to resort to violent means of committing suicide such as hanging, using firearms, wounding, VDI and wrist cutting. Some patients even adopted more than one method of suicide. Firearm (86%), wounding (67%) and hanging (73%) methods are more common among men (Table 4.8). Women were more likely to use VDI (64%) and wrist cutting (55%) (Table 4.8).

Note that 62% of those who were recruited were accompanied by someone to enroll in the Vig-ilanS programme (Tabel 4.2). Our data revealed that those that suicide recidivism was more common among those who were unaccompanied during their visit to enroll for the VigilanS programme.

The chi-squared test was performed to investigate the relationship between two qualitative variables. Statistical significance for the test was defined as p<0.05. Tabel 4.2 shows the predictors that have high association with recidivism. The chi-square test shows a strong dependence between recidivism and age with other essential variables such as alcohol consumption, which is a factor contributing to recidivism, and with other variables. recidivism in Nord-Pas-de-Calais. This model treats Recidivism after 6 months as the response variable, which informs the probability of the re-attempted suicide after 6 months of the current attempt. Male patients as well as those who were accompanied have lower probability of recidivism after 6 months. Missing hospitalisation after a violent suicide attempt increases the risk of reattempting suicide after 6 months.

One of the most prominent risk factors for death by suicide is a history of previous suicide attempts as a substantial number of patients who attempt suicide ultimately die committing suicide. If VigilanS can establish contact with the patient's family member, the chances of recidivism after 6 months are lower. The influence of more than one surrounding domain such as family, professional and social reduces the probability of recidivism after 6 months.

The risk of suicide recidivism is lower for those who were accompanied by someone such as family or friends to the hospital as well as those who were hospitalised. The severity of the previous suicidal act such as a violent act instils fear in the patient to reattempt suicide. Those patients with strong family support tend not to be victims of suicide recidivism.

When contact with a professional i.e. psychiatrist or psychologist is established, it indicates that the patient is in need of great help. Family and people around (entourage) play a vital role in giving support to the patient apart from consulting a professional to reduce suicide reattempts.

The duration of the VigilanS programme is 6 months and the decision of the clinician to prolong the duration indicates that the patients need more attention and require professional help. This also applies to the cases with follow-up after 10 days.

The spatial probit model, SAR The SAR and SEM models (see in the Appendix section 4.6 Tables 4.12 and 4.13) give similar results. The inclusion of spatial effects indicated by ρ (see Table 4.4) increased the AUC to 71.25%. The partially linear spatial probit model (PLSPM)

In attempting to further enhance the performance of the prediction model, features with nonlinear explanatory variables (exogenous determinants) were then added to the spatial probit regression model: median revenue (in Euros), unemployment rates, worker rates and graduation rates. We extract the smooth terms by fitting a GAM model (see Appendix Table 4.15) and use them as additive terms in the spatial probit model.

This study shows that significant predictors of suicide reattempts based on the best predictive Chapter 5

Concluding remarks and future research

Concluding remarks

The main theme of this thesis is spatial analysis. Spatial analysis can identify and solve complex location-oriented problems. Spatial analysis provides important insights in analysing contents of a map where characteristics of places and the relationships between these places are investigated.

This thesis is motivated by two real population health problems such as mortality modelling and the mapping of suicidal relapses. The first application deals with data of functional and spatial nature. Functional Data Analysis (FDA) is dedicated to analyze this kind of data.

FDA is an approach in statistics which encompasses the statistical methodology of data expressed in the form of functions. Data represented in the form of functions are then used in statistical modelling and prediction information can be retrieved from such data with the aid of some multivariable statistical concepts. In chapter 3 of this thesis, the study of the use of spatial statistics on functional data in demography was done. Mortality was investigated from a spatial perspective where the log of death rates data of 28 European countries were converted to functional data before performing spatial analysis on these data. The aim was to construct a tool to detect spatial autocorrelation in functional data. This led to the discovery of the functional Moran's I statistic. This statistic has the ability and potential to be used to detect spatial relationships involving functional data in various settings besides demography.

The second part of this thesis focuses on spatial modelling by considering a partially linear probit model for spatially dependent data. A deep dive on semi-parametric binary models from the theoretical perspective was conducted in Chapter 4. The combination of the generalized method of moments approach as well as the weighted likelihood method led to a semi-parametric estimation method. A spatial autoregressive error (SAE) model when the spatial dependence structure is integrated in a disturbance term of the studied model was addressed. This model was then applied to the suicide recidivism data collected by the VigilanS healthcare system in

2 p 2 ijCard

 22 -zero natural numbers: 1, 2 . . .Zset of integers: . . . , -1, 0, 1, . . .R set of real numbers: ] -∞, +∞[ R + set of real positives numbers: [0, +∞[ R d euclidian space of dimension d • integer part | • | absolute value if the argument is number or determinant if the argument is matrix • norm such that: if the argument is a vector x ∈ R d : x = x 2 1 + x 2 2 + • • • + x if the argument is a matrix A: A = a if the argument is a function f : f = sup |f (x)|x or x T transpose of vector or matrix x tr(•) trace of matrix⊗ Kronecker product Ā (or A c ) complement of set A A ∪ B union of A and B A ⊂ B A is included in B A ∩ B intersection of A and B A \ B set ofelements of A that are not included in B generated by (. . .) (Ω, A, P ) probability space. Ω : nonempty set, A :σ-algebra of subset of Ω and P : probability measure on A i.i.d independent and identically distributed N (0, 1) normal distribution

  the set of uncorrelated principal component scores, e t (x) is the residual function with mean zero and and K is the number of principal components used. The bivariate surface S t (x) -µ(x) is approximated through the sum of products of the orthonormal basis functions {φ k (x)} of age x and coefficients {β k (x)} of time t. The basis functions of equation (1.14) can be obtained by principal components and the uncorrelated coefficients are produced with PCA.

  .20) where I(•) is the indicator function; X and Z are explanatory random variables taking values in the two compact subsets X ⊂ R p (p ≥ 1) and Z ⊂ R d (d ≥ 1), respectively; the parameter β 0 is an unknown p × 1 vector that belongs to a compact subset Θ β ⊂ R p ; and g 0 (•) is an unknown smooth function valued in the space of functions G = g ∈ C 2 (Z) : g = sup z∈Z |g(z)| < C , with C 2 (Z) the space of twice differentiable functions from Z to R and C a positive constant. In model (3.1), β 0 and g 0 (•) are constant over i (and n). Assume that the disturbance term U in Chapter 1. State of art and general concepts in (3.2) is modelled by the following spatial autoregressive process (SAR):

Chapter 2 .

 2 Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data[START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] (HU) model extended the LC model by incorporating functional data analysis (FDA), nonparametric smoothing and robust statistics.[START_REF] Hyndman | Coherent mortality forecasting: The product-ratio method with functional time series models[END_REF] applied coherent forecasting of mortality rates for two or more subpopulations based on functional principal components models of simple and interpretable functions of rates to sex-specific data for Sweden and state-specific data for Australia.[START_REF] Greco | A space-time extension of the lee-carter model in a hierarchical bayesian[END_REF] combined mortality modelling techniques with the Bayesian approach in forecasting mortality rates by age and sex for provincial areas in Italy. Based on a study on county level mortality rates of the United States of America,[START_REF] Raymer | Spatial demography: A unifying core and agenda for further research[END_REF] argued that the mortality rate of a certain county may be associated with the features of its neighbouring counties beyond its own features. S.[START_REF] Wang | Spatial variations and macroeconomic determinants of life expectancy and mortality rate in china: A county-level study based on spatial analysis models[END_REF] explored the spatial distribution patterns and economic determinants of China by calculating the four indexes (lifespan expectancy at birth, infant mortality rate, under-5 mortality rate and crude mortality rate) at county level in China and illustrated the spatial distribution of these patterns.Principal component analysis (PCA) is of the essence in FDA. The combination of these elements is of great significance in enhancing spatial demographic modelling.[START_REF] Delicado | Statistics for spatial functional data[END_REF] provided a useful approach which involves the integration of classic types of spatial data structures (geostatistical data, point patterns, and areal data) with functional data. FDA approaches were applied to PCA in a spatial framework[START_REF] Li | Functional principal component analysis of spatiotemporal point processes with applications in disease surveillance[END_REF];[START_REF] Liu | Functional principal components analysis of spatially correlated data[END_REF]).[START_REF] Kuenzer | Principal component analysis of spatially indexed functions[END_REF] proposed a dimension reduction technique suitable for functional data, indexed by spatial locations on a grid. So far, there is no work on functional PCA (FPCA) on spatial data in the areal framework.

  neighbouring countries in Europe by employing a more specific form of principal component analysis developed to reduce multidimensionality in geo-referenced genetic data. This form, known as the spatial principal component analysis (sPCA) was introduced by[START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] and was formulated to investigate the spatial pattern of genetic variability using allelic frequency data of individuals or populations. sPCA is effective in revealing spatial connections in mortality data compared to the classical PCA. In our study, we consider 28 European countries whose mortality data are available on the Human Mortality Database (HMD) for a fixed period since it is of interest to investigate the existence of spatial autocorrelation in mortality data of neighbouring countries. The idea of sPCA by[START_REF] Jombart | Revealing cryptic spatial patterns in genetic variability by a new multivariate method[END_REF] aimed to reveal spatial patterns by directly considering spatial information. We constructed a functional Moran's

Figure 2 . 1 :

 21 Figure 2.1: The map of European countries with mortality data from HMD.

Figure 2 . 2 :

 22 Figure 2.2: The female log death rates in 2010 for 28 European countries: (a) observed, (b) smoothed, (c) reconstructed using the matrix multiplication of the scores and principal components based on contiguity weights, (d) reconstructed using the matrix multiplication of the scores and principal components based on KNN weights.

Figure 2 . 3 :

 23 Figure 2.3: The male log death rates in 2010 for 28 European countries: (a) observed, (b) smoothed, (c) reconstructed using the matrix multiplication of the scores and principal components based on contiguity weights, (d) reconstructed using the matrix multiplication of the scores and principal components based on KNN weights.

  Figures 2.2(b) and 2.3(b) represent the smoothed log death rates data using B-splines for the male and female populations in year 2010. The smoothed data for years 1990 and 2000 portray similar characteristics to those in Figures 2.2(b) and 2.3(b).

  Figure 2.4 illustrates the smoothed functional Moran statistics for ages 0 to 100+ using Bsplines for the female and male log death rates data based on the KNN and contiguity weights.

Figure 2 .

 2 Figure 2.4 also highlights that the functional Moran's I statistics showed comparable behaviour for the KNN and contiguity weights for all three years. For females, higher spatial dependency is visible from approximately 20 years of age to 80 years of age. The similar attributes are observed in the log death rates of males. Generally, for ages approximately 80 years and above, spatial autocorrelation decreases because the number of people who live past 80 is low. The life expectancy for the countries in Europe is between 70 to 80 years of age, from 1990 to 2010 (European Union Data, 2020). This contributed to the high spatial dependency (approximately

Chapter 2 .Figure 2 . 4 :

 224 Figure 2.4: Smoothed functional Moran's I statistics curves for log death rates from ages 0 to 100+ of: (a) females using KNN, (b) females using contiguity, (c) males using KNN, and (d) males using contiguity, weight matrices for years 1990, 2000 and 2010.

Figure 2 .

 2 Figure 2.7 displays the projection of the log death rates scores for females in year 2010 using the KNN neighbours of the three spatial PCs (top two positive and top one negative PC scores) onto the geographical map. The first positive PC (Figure 2.7(a)) shows spatial connectivity between the states split into two clusters, one in the west and one in the east. The projection for the second positive PC (Figure 2.7(b)) also shows spatial connectivity where two clusters are formed (northern and southern regions). The first negative PC (Figure 2.7(c)) does not seem to display a particular spatial pattern. Similar patterns are observed for cases involving the contiguity neighbours (Figure 2.8). This outcome is anticipated because the negative principal components are associated with local structures that highlight dissimilarities on the geographical map at neighbouring locations. In general, spatial patterns are noticeable for the first and second PCs involving male and female mortality data for years 1990, 2000 and 2010. These characteristics appear to be similar for the KNN and contiguity neighbours.

Figure 2 . 5 :Figure 2 . 6 :

 2526 Figure 2.5: Eigenvalues based on FASPCA using the KNN weight matrices for (a) females and (b) males from 28 European countries in 2010.

Figure 2 . 7 :Figure 2 . 8 :

 2728 Figure 2.7: The scores of the (a) first positive, (b) second positive, and (c) first negative eigenvalues of the FASPCA based on KNN weights for females from 28 European countries in 2010.

Figure 2 . 9 :

 29 Figure 2.9: The scores of the first eigenvalues based on the classical FPCA for (a) females and (b) males from 28 European countries in 2010.

Figure 2 .Figure 2 .

 22 Figure 2.10: The scores of the (a) first positive, (b) second positive, and (c) first negative eigenvalues based on KNN weights for males from 28 European countries in 2010.

  .2) where I(•) is the indicator function; X and Z are explanatory random variables taking values in the two compact subsets X ⊂ R p (p ≥ 1) and Z ⊂ R d (d ≥ 1), respectively; the parameter β 0 is an unknown p × 1 vector that belongs to a compact subset Θ β ⊂ R p ; and g 0 (•) is an unknown smooth function valued in the space of functions G = g ∈ C 2 (Z) : g = sup z∈Z |g(z)| < C , with C 2 (Z) the space of twice differentiable functions from Z to R and C a positive constant.

  .14) where K(•) is a kernel from R d to R + and b n is a bandwidth depending on n. Now, replacing g θ (•) in (3.11) by the estimator ĝθ (•) permits one to obtain the GMM estimator θ of θ as θ = argmin θ∈Θ Q n (θ, ĝθ ). (3.15) A classical inconvenience of the estimator ĝθ (z) proposed in (3.14) is that the bias of ĝθ (z) is high for z near the boundary of Z. Of course, this bias will affect the estimator of θ given in (3.15) when some of the observations Z in are near the boundary of Z. A local linear method,

Theorem 3 . 2 . 1 .

 321 Under Assumptions A1-A10, we have θ -θ 0 = o p (1). Corollary 3.2.1. If the assumptions of Theorem 3.2.1 are satisfied, then we have ĝθ -g 0 = o p (1).

Figure 3 . 1 :

 31 Figure 3.1: The true function g(•) and the average of its estimates, over the 300 replications

  j d n (θ, z) = O p (1), j = 0, 1, 2.

  .52) Let N 0 be an open neighbourhood of θ 0 , and consider the compact set Θ 0 = Θ \ N 0 . Let {N θ : θ ∈ Θ, θ = θ 0 } denote the open covering of Θ 0 by the procedure given above (each neighbourhood N θ satisfies (3.51))

  Ũin ∂η (θ, η) = O p (1), 1 ≤ i ≤ n, n = 1, 2, . . . (3.59)

ξ

  .64) Then, ∂G in (θ,η) ∂θ is bounded uniformly in i, n, θ, η by Assumptions A6 and A9 and the compactness of Θ (see assumption A7). This completes the proof of (3.58); hence, (3.56) is proved.Lemma 3.5.5. Under the assumptions of Proposition 3.5.1 and Assumptions A6 and A9, we havesup θ∈Θ S n (θ, ĝθ ) -S n (θ, g θ ) = o p (1). (3.65)If in addition M n -M = o p (1), then we havesup θ∈Θ |Q n (θ, ĝθ ) -Q n (θ, g θ )| = o p (1).Let us prove(3.65). For each θ ∈ ΘS n (θ, ĝθ ) -S n (θ, g θ ) = n -1 n i=1 ξ i Ũin (θ, ĝθ ) -Ũin (θ, g θ ) in Ũin (θ, ĝθ ) -Ũi (θ, g θ ) because sup i,n ξ in = O p(1) (by Assumption A6), sup θ ĝθ -g θ = o p (1) (see Proposition 3.5.1) and sup θ,η ∂ Ũin ∂η (θ, η) = O p (1) uniformly on i and n (see the proof of Lemma 3.5.4). The proof of (3.66) is made trivial by combining (3.65) with Assumption A10.

  .69)We remark that(3.68) is equivalent tosup θ d dθ S n (θ, ĝθ ) -d dθ S n (θ, g θ ) = o p (1) T S n (θ, ĝθ ) -d 2 dθdθ T S n (θ, g θ ) = o p (1)(3.71) by (3.11) (because M n -M = o p (1) thanks to Assumption A10) and sup θ S n (θ, ĝθ ) -S n (θ, g θ ) = o p (1)

  on i and n, and θ -θ 0 = o p (1).

  .83) Consider the second term on the right-hand side in (3.83), where we remark that because ∂S ∂g (θ 0 , g 0 ) and sup θ sup z ∂g θ (z) ∂θ∂θ T are finite and θ -θ 0

  j (ĝ θ -g θ ) are all of order o p (1) by Proposition 3.5.1, it suffices to show that sup θ,η sup i ∂ Ũin ∂η (θ, η) = O p (1) η) = O p (1). (3.90) Equation (3.89) is already proved in the proof of Lemma 3.5.4 (see (3.59)). The proof of (3.90) Chapter 3. Partially linear spatial probit models can be established in a similar manner and is thus omitted.

Figure 4

 4 Figure 4.1: Map of suicide attempts per 1000 people

  and follow the partially linear model of a latent dependent variable Y * :Y * in = X T in β 0 + g 0 (Z in ) + U in , 1 ≤ i ≤ n, n = 1, 2, . . .(4.3)withY in = I (Y * in ≥ 0) , 1 ≤ i ≤ n, n = 1, 2, . . . (4.4) where I(•) is the indicator function; X and Z are explanatory random variables taking values in the two compact subsets X ⊂ R p (p ≥ 1) and Z ⊂ R d (d ≥ 1), respectively; the parameter β 0 is an unknown p × 1 vector that belongs to a compact subset Θ β ⊂ R p ; and g 0 (•) is an unknown smooth function valued in the space of functions G = g ∈ C 2 (Z) : g = sup z∈Z |g(z)| < C , with C 2 (Z) the space of twice differentiable functions from Z to R and C a positive constant.In model(4.3), β 0 and g 0 (•) are constant over i (and n). Assume that the disturbance term U in in (4.4) is modelled by the following spatial autoregressive process (SAR):U in = λ 0 n j=1 W ijn U jn + ε in , 1 ≤ i ≤ n, n = 1, 2, . . . (4.5)where λ 0 is the autoregressive parameter, valued in the compact subset Θ λ ⊂ R, W ijn , j = 1, . . . , n are the elements in the i-th row of a non-stochastic n × n spatial weight matrix W n .

A

  Moran's I test was conducted on the residuals for the aforesaid model to investigate the existence of spatial autocorrelation based on spatial weight matrix created from the location coordinates of the address of each patient. Since the Moran's I test appeared to be significant with I = 0.1497 and p-value = 0.001, spatial effects were included to the probit regression model.The p-value of ρ suggests that spatial effects are significant in this model and the AUC increases to 71.25%.

Chapter 4 .

 4 Application of spatial models to investigate suicide recidivism in Nord-Pas-de-Calais.

  model i.e the partially linear spatial probit model after a 6-month follow-up are gender, total number of suicide attempts by patient, if patient was accompanied to hospital, hospitalisation, contact with family, friends or professionals (psychiatrists), 10-day psychological monitoring, 3impact domain (affected by family, social and professional surroundings), monitoring of patient for 6-months, patients with unidentified native countries, and finally the combination of violent methods of suicide attempt as well as hospitalisation. The Moran's I test for each of these variables showed the existence of spatial autocorrelation (Table 4.5). The spatial component for this model is undoubtedly significant (p-value of ρ) resembling the earlier model. All the other predictors used are similar to the earlier model. This partially linear spatial probit model has an AUC of 79.76%. The AUC curve for this model is given in Figure 4.2a. A binned residual plot (Figure 4.2b) was also constructed for the partially linear spatial model to assess the fit of this model. Only a few points lie outside the confidence limits but no systematic pattern is detected in the plot. Therefore, this plot indicates that the partially linear spatial probit model is a good fit to the data compared to some of the investigated models from the probit regression model paradigm. (a) (b)

Figure 4 . 2 :

 42 Figure 4.2: Binned and AUC plots of PLSPM

Figure 4 . 8 :

 48 Figure 4.7: Smoothed terms

France.

  It was interesting to observe the existence of spatial autocorrelation of suicide recidivism cases in Nord Pas de Calais. The partially linear spatial autocorrelation model proved to be a good fit in modelling suicide recidivism by considering spatial autocorrelation and partially linear functions which were obtained based on potential exogenous determinants of suicide such as median income, graduation rates, unemployment rates and worker rates of populations in the

  Mortality, fertility, and migration are processes which can change the population. The twentieth century witnessed the improvement in mortality especially in developed countries. Mortality forecasting has been gaining interest and this led to the development of stochastic mortality forecasting approaches over time which involve extrapolation and time series methods. The Lee-

	2. Mortality modeling	5
	1.2 Mortality modeling
	1.2.1 The Lee-Carter model
	Carter (LC) model

  Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data For weights based on boundaries, we often use spatial contiguity to specify neighbouring locations in the sense of sharing a common border. There are various types of spatial contiguities but the classical cases are those known as the Rook's contiguity (where two cells of a matrix which share a common boundary are neighbours), the Bishop's contiguity (where two cells of a matrix share a common vertice) and the Queen's contiguity (neighbours by either the Rook's or the Bishop's contiguity). The contiguity weights are given as Wilk test to assess multivariate normality on the log of death rates data is required to investigate if the data fulfills the normality assumption. If the data fulfils this assumption, we can proceed with computing the functional Moran's I statistic for the KNN-and contiguitybased neighbourhoods to show spatial autocorrelation of log death rates. If the data violates the normality assumption, we are required to run a Monte Carlo simulation on the Moran's I

	w ij =	1 if i and j are contiguous; 0 otherwise.
	A Shapiro-

statistic. It is essential to investigate the global autocorrelation which measures the degree of clustering as well as the local indicators which allows the decomposition of the Moran's I global indicator into the contribution of each observation. The spdep (Bivand & Wong, 2018) package in the R software was used to aid the use of spatial weights as well as the Moran's I statistic to measure the spatial autocorrelation. For the first part of the study, the classical functional PCA is performed on the smoothed data (Figures 2.2(b) and 2.3(b)). The classical functional PCA does not account for spatial information. Hence, the application of FASPCA enhanced with the implementation of the multivariate (non-functional) spatial pca (sPCA) by

  table, a classical Moran's I index is calculated considering for each year and gender, the raw data matrix as a panel dataset. Positive spatial autocorrelation indicates that locations nearby tend to be similar on map where high values tend to be near high values and low values near low values. On the other hand, when geographic values are dissimilar, the map shows negative spatial autocorrelation. The Moran's I statistics reported values close to +1 for both males and females in years 1990, 2000 and 2010. This suggests that neighbouring locations have strong positive autocorrelation in mortality for both KNN and contiguity neighbours. The Moran's I index reported in Table 1 are somehow aggregations of the functional index (equa-

tion

(2.4)

) like the functional trace index defined in equation (2.7).

Table 2

 2

.1: Moran's test for spatial autocorrelation based on log of death rates for females and males in 28 countries in Europe based on KNN and contiguity weights.

Table 2 . 2 :

 22 Moran's test on principal components using FPCA and FASPCA, based on KNN and contiguity weight matrices for females and males of 28 European countries in 2010.

		Female		Male
		Moran's I variability (%) Moran's I variability (%)
	Classical FPCA
	1 st score	0.4973 * * *	99.81	0.5482 * * *	99.67
	2 nd score	0.0402	0.08	-0.0873	0.14
	3 rd score	0.2305 * *	0.04	0.2687 * *	0.08
	4 th score	0.1448 *	0.03	-0.0548	0.04
	Total		99.96		99.93
	FASPCA (KNN (3,2))
	1 st score positive	0.5400 * * *	77.74	0.6021 * * *	85.14
	2 nd score positive	0.3442 * *	8.93	0.3239 * *	2.54
	3 rd score positive	0.1636 *	4.09	0.1821 *	0.86
	2 nd score negative	-0.1775 *	2.29	-0.1575 †	2.29
	1 st score negative	-0.1030	3.72	-0.1516 †	6.65
	Total		96.77		97.48
	FASPCA (Contiguity (3,2))
	1 st score positive	0.5041 * *	67.15	0.6315 * * *	83.41
	2 nd score positive	0.3857 * *	14.37	0.1831 †	1.87
	3 rd score positive	0.3890 * *	5.89	0.2537 *	1.78
	2 nd score negative	-0.1546	1.64	-0.2023	1.60
	1 st score negative	-0.3316 *	5.99	-0.2544 †	8.61
	Total		95.04		97.27

Note: † p < 0.1; * p < 0.05; * * p < 0.01; * * * p < 0.001

Table 2 .

 2 3: Moran's test on principal components using FPCA and FASPCA, using KNN and contiguity weight matrices for females and males of 28 European countries for 1990.

		Female		Male
		Moran's I variability (%) Moran's I variability (%)
	Classical FPCA
	1 st score	0.4135 * * *	99.79	0.5233 * * *	99.73
	2 nd score	-0.1203	0.08	-0.1342 †	0.11
	3 rd score	-0.0019	0.05	0.3420 * * *	0.06
	4 th score	0.4431 * * *	0.03	-0.0271	0.03
	Total		99.95		99.93
	FASPCA (KNN (3,2))
	1 st score positive	0.4905 * * *	65.46	0.5527 * * *	81.18
	2 nd score positive	0.4211 * *	11.77	0.3575 * * *	5.33
	3 rd score positive	0.1067 †	1.64	0.0957 †	0.88
	2 nd score negative	-0.1026	4.51	-0.2093 * *	2.98
	1 st score negative	-0.1324	10.66	-0.1655 *	7.78
	Total		94.04		98.15
	FASPCA (Contiguity (3,2))
	1 st score positive	0.5723 * * *	58.16	0.6376 * * *	80.64
	2 nd score positive	0.5975 * * *	13.96	0.3917 *	6.38
	3 rd score positive	0.2018 †	6.52	0.2248 †	2.90
	2 nd score negative	-0.0948	2.26	-0.1105	2.32
	1 st score negative	-0.3174 *	12.30	-0.2637 †	3.71
	Total		93.2		95.95

Note: † p < 0.1; * p < 0.05; * * p < 0.01; * * * p < 0.001

Table 2 .

 2 4: Moran's test on principal components using FPCA and FASPCA, using KNN and contiguity weight matrices for females and males of 28 European countries for 2000.Chapter 2. Exploring spatial patterns of mortality in Europe using functional spatial principal components for areal data

		Female		Male
		Moran's I variability (%) Moran's I variability (%)
	Classical FPCA
	1 st score	0.4710 * * *	99.75	0.5418 * * *	99.69
	2 nd score	-0.1141	0.12	0.1414 *	0.15
	3 rd score	0.0070	0.06	0.4498 * * *	0.07
	4 th score	0.3234 * * *	0.02	-0.1972 *	0.03
	Total		99.95		99.94
	FASPCA (KNN (3,2))
	1 st score positive	0.4786 * * *	71.66	0.5645 * * *	83.74
	2 nd score positive	0.4584 * * *	6.79	0.2613 * *	9.24
	3 rd score positive	0.0968 †	1.69	0.1128 †	0.66
	2 nd score negative	-0.1553 *	3.94	-0.1790 *	1.78
	1 st score negative	-0.2072 *	13.27	-0.1563 *	3.10
	Total		97.35		98.52
	FASPCA (Contiguity (3,2))
	1 st score positive	0.5182 * *	72.7	0.6758 * * *	87.16
	2 nd score positive	0.3055 *	7.53	0.1626	5.56
	3 rd score positive	0.2524 *	3.78	0.2606 *	1.91
	2 nd score negative	-0.1558 †	4.32	-0.0977	1.11
	1 st score negative	-0.1479	6.30	-0.3445 *	1.94
	Total		94.63		97.68

Note: † p < 0.1; * p < 0.05; * * p < 0.01; * * * p < 0.001

  .17) Assumption A2. (Local dependence). The density f in (•) of Z in exists, is continuous on Z uniformly on i and n and satisfies

	lim inf n→∞	inf z∈Z	1 n	n i=1

  VDI), phlebotomy, jumping from a great height, combination of several methods and others); duration of hospitalisation; information concerning calls (answered and missed calls, and the number of these calls) that will be made on the 10th day as well as 6 months after the suicide attempt; and if there has been contact with the patient's family and friends or a profes-

	4.2. Data description	71
	of drugs, drug poisoning (medicaments), drowning, wounding (lesions), hanging, voluntary drug
	intoxication (sional.
	The data at day 10 is based on the responses of the patients to the questionnaire on the 10th
	day. It contains variables such as the date of call, the duration of call, and the responses (the
	options are in parentheses) to the following questions (classified as categorical variables):
	1. how does the patient feel? (Better, Worse, No changes)
	2. does support from family, professional (psychiatrist) and social domain help to overcome
	the temptation to commit suicide? (Yes/ No)
	3. is there psychological monitoring of patient? (Yes/ No)
	4. does the patient need help? (Yes/ No)
	5. is the patient suffering and needs help? (Suffering and needs help/ suffering but does not
	need help/ in difficulty/ no issues)
	6. effect of postcard intervention (Helpful/ Not helpful)
	7. available support (Friends, family, VigilanS professional, attending psychiatrist, doctor,
	Psychiatric Medical Centre (inpatient treatment), Hospital Emergency (outpatient treat-
	ment), Others)
	On the other hand, the 6-month data is based on the patient's responses to the 6-month
	questionnaire. The variables are related to the date and duration of the call and the patient's
	responses (the options are in parentheses) to the following questions:
	1. display of risky behaviour since the attempt (Yes/ No);
	2. existence of suicidal thoughts since the attempt (Yes/ No);
	3. continuous monitoring on patients (Yes/ No);
	4. effect of postcard intervention (Helpful/ Not helpful);
	5. the total number of lifetime suicide attempts.

This 6-month follow up study collected data at the point of recruitment, 10 days after and 6 months after recruitment. The database includes the data of more than 23000 patients from different establishments, with information on sociodemographic characteristics such as date of birth (age), gender, address, and native country. This data set also contains information on alcohol consumption before the suicide attempt; history of suicide attempts; suicide methods (overdose

Table 4 .

 4 2: Factors of suicide recidivism and methods used. Chapter 4. Application of spatial models to investigate suicide recidivism in Nord-Pas-de-Calais. This model resulted in an AUC of 69.54%. It was of interest to study if spatial effects are of significance to predicting recidivism.

	Variables	Total Sample recidivism N= 1364 N= 417	No recidivism t/χ 2 df N=947	p-value
	Gender				3.53		0.06
	Male (%)	433 (31.74)	117 (28.06)	316 (33.37)
	Female (%)	931 (68.26%) 300 (71.94)	631 (66.63)
	Age (mean±SD)	42.58 ± 15.38 42.87 ± 13.35 42.45 ± 16.19 -0.51 953.42 0.6
	Alcohol (Yes)	689 (50.51)	212 (50.83)	477 (50.37)	0.01	1	0.8
	Companion (Yes)	984 (62.14)	275 (65.95)	709	11.02 1	0.0008
	TotalSA	7.396 ± 13.59 9.66 ± 16.17	6.4 ± 12.15	-3.68 631.43 0.00025
	Hospitalisation (Yes) 593	172	421	1.08	1	0.2
	Methods				36.10 8	1.68 e -5
	Firearm	5	0	5
	Other ways	24	0	24
	VDI	1075	334	741
	Wounding	12	2	10
	Drowning	11	4	7
	Hanging	31	3	28
	Phlebo	112	52	60
	Several ways	82	19	63
	Jump	12	3	9
	Native country				30.4	2	2.5e -7
	Not France	32	11	21
	France	1047	357	690
	Not available	285	49	236
	Contact				30.11 3	1.3e -6
	Entourage	126	20	106
	Professional	320	121	199
	Ent/prof	259	97	162
	No contact	659	179	480
	Impact domain				8.88	3	0.03
	One domain	535	179	356
	Two domains	209	59	150
	Three domains	40	5	35
	No impact	580	174	406

Table 4 .

 4 4: Results for the SAR probit model

	Predictors	Estimate Std Error p-value
	Intercept	-0.6567	12.3006 4.53e -04 ***
	Age	0.0032	2.1130	0.1460
	Gender (Male)	-0.0621	0.9706	0.3245
	TotalSA	0.0054	2.9662	0.0850 †
	Alcohol (Yes)	0.0981	0.4347	0.5097
	Companion (Yes)	-0.1626	4.5991	0.0320 *
	VSA (Yes)	-0.4942	16.2366 5.59e -05 ***
	Hospitalisation (No)	-0.1013	2.3747	0.1233
	Contact (Ent & Prof)	0.1926	5.4671	0.0194 *
	Contact (Entourage)	-0.3487	3.5624	0.0591 †
	Contact (Professional)	0.1081	5.0848	0.0241 *
	One impact domain	0.0698	0.2995	0.5842
	Two impact domains	-0.0104	0.1127	0.7371
	Three impact domains	-0.3460	2.7501	0.0972 †
	Follow up after 6 months (Yes)	0.2819	10.0414	0.0015 **
	Psychological monitoring D10 (Yes)	0.4377	24.1588 8.87e -07 ***
	Native country (Not Available)	-0.4843	25.6858 4.02e -07 ***
	Native country (Not.France)	0.1162	0.0810	0.7759
	VSA (Yes): Hospitalisation (No)	0.6192	18.5235 1.68e -07 ***
	ρ	0.4524	73.7243 8.98e -18 ***
	AUC	0.7125
	AIC	1558.7180

Note: † p < 0.1; * p < 0.05; * * p < 0.01; * * * p < 0.001

Table 4

 4

	.5: The Moran's test
	Variables	Moran's I
	Median revenue	0.2048 ***
	Unemployment rate	0.1598 ***
	Worker rate	0.3028 ***
	Graduation rate	0.3457 ***
	1 st score PCA	0.2476 ***
	1 st score positive (sPCA) 0.2048 ***

Note: *** p < 0.001

Table 4 .

 4 13: The SEM model after 6 months.Note: † p < 0.1; * p < 0.05; * * p < 0.01; * * * p < 0.0014.6. Appendix

	Variable	Estimate	Std Error	p-value
	Intercept	-0.897245842	16.14462191	5.87e -05	***
	Age	0.003935661	1.40424956	2.36e -01
	Gender(male)	-0.096120316	0.2099344	6.47e -01
	TotalSA	0.006516742	2.01090302	1.56e -01
	Alcohol(Yes)	0.037510691	0.07331228	7.87e -01
	Companion(Yes)	-0.241803841	7.21854051	7.22e -03	***
	TS.Violent(Yes)	-0.632613562	18.48920955	1.71e -05	***
	Hospitalisation(No)	-0.117829442	2.34092106	1.26e -01
	Contact (Entourage & Professional)	0.288350319	9.02851656	2.66e -03	**
	Contact (Entourage)	-0.310575309	2.61167453	1.06e -01
	Contact(Professional)	0.151338502	3.17268511	7.49e -02	†
	Impact.Domains (1)	0.094252792	0.70474297	4.01e -01
	Impact.Domains(2)	-0.044340637	0.26630705	6.06e -01
	Impact.Domains (3)	-0.561442638	4.93141408	2.64e -02	*
	Follow up after 6 months (Yes)	0.35109305	9.28905535	2.31e -03	**
	Psychological monitoring D10 (Yes)	0.460645284	17.51353636	2.85e -05	***
	Native.region(Not Available)	-0.597281804	24.50062316	7.43e -07	***
	Native.country(Not.France)	0.082105223	0.10788625	7.43e -01
	TS.Violent(Yes):Hospitalised(NH)	0.889339097	24.20113291	8.68e -07	***
	rho	0.528275364	76.80227304	1.89e -18	***
	AUC	0.6955

Acknowledgments

However, a Thank You is only appreciable if it is really sincere, so I will try to choose the right words to thank my God who gave me the chance to meet more than remarkable people during my life.

We have (3.30) because ζ θ (Y in ) is bounded uniformly on i and θ by assumption H.1, f i (z -ub)du ≤ C (see assumption A2) and sup u |K(u)| 2 < ∞ (see Assumption A4 and the compactness of Z). Then, we have

Now, consider the covariance term

Let us partition the spatial locations of the observations using

with c n being the sequence of integers going to ∞, and let Dn denote the complement of D n in the set of locations {s i , i = 1, ..., n}.

On the one hand, let

by Assumption H.1, sup u |K(u)| < ∞ (Assumption A4 and the compactness of Z), with f i,j

being the joint density (Assumption A2 and the compactness of Z).

Note that the second term B is

Chapter 3. Partially linear spatial probit models

Consequently, we obtain

∂S ∂θ (θ 0 , g 0 ) + ∂S ∂g (θ 0 , g 0 )g 0 + o p (1).

Then, we have θ -θ 0 = -{B 2 (θ 0 )} -1 ∂S ∂θ (θ 0 , g 0 ) + ∂S ∂g (θ 0 , g 0 )g 0 T M S n (θ 0 , g 0 ) + o p (1).

To end the proof, it remains to be shown that √ nB 1 (θ 0 ) -1/2 S n (θ 0 , g 0 ) -→ N (0, I q ).

Consider, for all w ∈ R q such that w = 1,

By the Cramer-Wold device, it suffices to show that A n converges asymptotically to a standard normal distribution, for all w ∈ R q , such that w = 1.

To prove this, we will use the central theorem limit (CTL) proposed by [START_REF] Pinkse | A central limit theorem for endogenous locations and complex spatial interactions[END_REF].

These authors used an idea of [START_REF] Bernstein | Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes[END_REF] based on partitioning the observations into J groups G n1 , . . . , G nJ , 1 ≤ J < ∞, which are divided up into mutually exclusive subgroups G j1n , . . . , G jm jn n , j = 1, . . . , J. Each observation belongs to one subgroup, and its membership can vary with the sample size n, as can the number of subgroups m jn in group j. We assume that the partition is constructed such that

Partial sums over elements in groups and subgroups are denoted by A nj and A jtn ,j = 1, . . . , J, and t = 1, . . . , m jn , respectively. Thus, we have

Let us recall in the following the assumptions under which the CTL of [START_REF] Pinkse | A central limit theorem for endogenous locations and complex spatial interactions[END_REF] holds.

Assumption A. For any j = 1, . . . , J, let G * , G * * ⊂ G jn be any sets for which

Then, for any function

, where ι is Chapter 4. Application of spatial models to investigate suicide recidivism in Nord-Pas-de-Calais.

• if REC = Yes and duration is more than 6 months, REC6 = Yes,

• if REC = Yes and duration is lesser than 6 months, REC6 = No,

• if REC = No, then REC6 = No.

Data for exogenous determinants

Apart from the data collected by VigilanS, it was of interest to examine the impact of exogenous determinants such as the income, the manual worker rate and the level of education of the population in the Nord-Pas-de-Calais region. The data was retrieved from website atlasante.fr.

Predictors examined

The predictors examined at initial point of recruitment and 6 months after recruitment are given in the following table:

The probit regression model

The classical probit regression model (equation 4.2) predicting possible recidivism 6 months after the current attempt was validated.

PLSPM with deprivation index

We calculate the index of deprivation for each commune by extracting the first component of the principal component analysis (PCA) of 4 available variables namely, worker rate, graduation rate, unemployment rate and median revenue. We conduct spatial PCA and the first positive component is also used as a spatial index of deprivation. The Moran's test on these components is well significant (Table 4.5). From the GAM model (Table 4.16 in Appendix), we use this index as an additive term in this spatial probit model (Table 4.7).

Conclusion

The application of the partially linear spatial probit model on the suicide recidivism data clearly showed that the inclusion of the spatial effect significantly improved the prediction ability of the model. This model also identified significant predictors of suicide recidivism as well as exogenous determinants which can contribute towards suicide reattempts. Multiple suicide attempters must be given more attention and the VigilanS programme is working towards significantly lowering suicide cases in Northern France. This statistical analysis can work hand in hand with the recidivism in Nord-Pas-de-Calais.

VigilanS programme towards achieving its goal. Socio-familial isolation is considered a fairly important risk factor of suicide recidivism attempt. Moreover, a patient committing a violent act of suicide and was not hospitalised tends to repeat the act.

Appendix

Future research

Future work in the field of demography can shed some light on constructing spatial predictive models to predict mortality rates for neighbouring countries with limited or no data. This idea can further be extended to the spatio-temporal framework where mortality of a certain location can be predicted at a certain time.

The second part of the thesis and its application in investigating suicide recidivism, the choice of the instrumental variables for a more efficient regression estimator is a hot point in particular for real data.

A future work related to the project entitled: "Impact of the COVID-19 pandemic context on suicidal behaviors and their management", proposes to study the impact of the pandemic context on the incidence of suicide attempts and deaths by suicide and on suicidal ideation (based on content published on social networks) during the different periods that marked the pandemic (over a period of 3 years beginning 1 year before the pandemic).

A population which is vulnerable to suicidal behaviour i.e. that with high number of suicide history will be investigated. Finally, as the pandemic context is likely to influence the management of suicidal behaviour, a qualitative exploration of the difficulties encountered will be carried out.

The consequences of the COVID-19 pandemic and its associated measures on the prevalence of psychiatric troubles but also on the key factors associated with suicide (e.g. social isolation, precariousness) as well as the reduction of access to psychiatric care during the health crisis indicate the fear of increased suicidal behaviors which has not been evaluated in France. The objectives of the project include:

• quantitative investigation of the impact of the Covid19 pandemic and health measures on suicidal behaviours (suicide attempts and deaths by suicide) in the general population, and in a particularly vulnerable population (people with a history of suicide attempts);

• extensively study the content posted on social networks to examine the impact of the pandemic and health measures on suicidal thoughts;

• qualitatively analyse the difficulties of dealing with attempted suicides in the pandemic context.

The first part of the analysis will concern VigilanS activity data for the population of four French French territory and in accordance to time. This will allow us to study the spatial changes in the incidence of suicide attempts and deaths by suicide before, during and after the pandemic period.

In order to study the impact of territorial factors related to the pandemic context by taking into account the proven risk factors of suicidal behaviors, we will employ recurrent multivariate models (by taking geography into account), multilevel models and spatial econometric and statistical specifications to map ecological risks. Moreover, joint modeling of individual and ecological risk factors avoid problems of ecological and individual inference and thus substantially reduce uncertainty about the interpretation of the respective role of each level of risk factors.