Imen Ouled Dlala

Dorra Attiaoui

Arnaud Martin

Boutheina Ben Yaghlane

Trolls Identification within an Uncertain Framework

Keywords: Q&AC, trolls, belief function theory, conflict measure

The web plays an important role in people's social lives since the emergence of Web 2.0. It facilitates the interaction between users, gives them the possibility to freely interact, share and collaborate through social networks, online communities forums, blogs, wikis and other online collaborative media. However, an other side of the web is negatively taken such as posting inflammatory messages. Thus, when dealing with the online communities forums, the managers seek to always enhance the performance of such platforms. In fact, to keep the serenity and prohibit the disturbance of the normal atmosphere, managers always try to novice users against these malicious persons by posting such message (DO NOT FEED TROLLS). But, this kind of warning is not enough to reduce this phenomenon. In this context we propose a new approach for detecting malicious people also called 'Trolls' in order to allow community managers to take their ability to post online. To be more realistic, our proposal is defined within an uncertain framework. Based on the assumption consisting on the trolls' integration in the successful discussion threads, we try to detect the presence of such malicious users. Indeed, this method is based on a conflict measure of the belief function theory applied between the different messages of the thread. In order to show the feasibility and the result of our approach, we test it in different simulated data.

I. INTRODUCTION

T HE way we look for, and acquire information has shifted greatly into to instant, easy and low cost process. In fact, thanks to the Internet one can make a research in any given topic, get a huge amount of information by a simple click. Although, for some problems it is difficult to get satisfactory answers by searching directly on a traditional search engine. Instead, we prefer to find someone who has expertise or experience. In order to have the best answer, one of the tools that has widened the scope of information exchange is Question Answering Communities (Q&AC). These systems allow everyone to contribute as much as they can on a given community. Unfortunately, not all messages can be considered as reliable: some users claim themselves as experts, and other people post messages without any utility for the one who is seeking for answers. Thus, the managers of these communities seek to always enhance the performance of such platforms. Although, the increase of the useless messages can be attributed to the presence of trolls. The term of trolling has been defined in several works within different communities, including [START_REF] Hardarker | Trolling is asynchronous computer-mediated communication: from user discussion to academic definitions[END_REF], [START_REF] Bouguessa | Identifying authoritative actors in question answering forums: the case of Yahoo! answers[END_REF] and [START_REF] Mcqueen | Some methods for classification and analysis of multivariate observations[END_REF]. These Malicious people intend to insidiously mislead the subject of the discussion in order to provoke controversy and disrupt the discussion. They aim to make normal users fall into their traps by deviating them from the main topic of the discussion. In fact, the only way to deal with a troll is to ignore him, or detect his presence in order to notify him or take away his ability to post online. Thus, Manuscript received. some other works tried to detect not just their characteristics but also their presence in order to avoid them. To address this problem Cambria et al. [START_REF] Patxi | Supervised machine learning for the detection of troll profiles in twitter social network: Application to a real case of cyberbullying[END_REF] proposed a technique based on semantic computing to automatically detect and check web trolls. This work aims to prevent the malicious people from emotionally hurting other users or communities within the same social network. In another work Ortega et al. [START_REF] Shachaf | Beyond vandalism: Wikipedia trolls[END_REF] proposed a method to classify users in a social network regarding to their trustworthiness. The goal of their method is to detect trolls from the other users by preventing such malicious users to gain high reputation in the network. Patxi et al. [START_REF] Dubois | A set theoretic view of belief functions: logical operation and approximations by sets[END_REF] dealt with Trolling users on twitter social network. These studies were explored in different social networks within certain framework.

When dealing with real-world applications, the massive amounts of data are inseparably connected with imperfection. In fact, this kind of data can be imprecise and/or uncertain or even missing. Different theories have emerged to deal with this kind of data such as fuzzy set theory [21], possibility theory [22] and belief function theory [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF]. Thus, to be closer to reality and to obtain more relevant results, we propose a new method dealing with uncertain data. This method aims to detect trolls in Q&AC using the framework of belief function.

The paper proceeds as follows: in Section 2, we introduce the Q&AC and briefly review related works. In section 3, we present the necessary background regarding the different concepts of the belief functions theory. We define the different steps of our proposal based on a conflict measure in section 4. Finally, we present the feasibility of the proposed method on an illustrated example.

II. Q&AC: QUICK OVERVIEW

In this section we introduce some concepts related the Q&AC. First we will start by presenting the main actors in these forums, then a little overview on sources identification and finally the levels of uncertainty we can face in Q&AC.

A. Users within Q&AC

Users are considered as the main actors within Question Answering Communities. We can define different types such as experts, trolls and learners.

• Reliable user / Expert: a person who is very knowledgeable about or skilful in a particular area. • Troll: a person who seeks to disturb the serenity of the concerned community. His purpose is to create controversial debates by multiplying irrelevant messages that we keep unanswered. • Learner: a normal user of the Question Answering

Community, trying to gain information and expertise.

B. Sources Identification within Q&AC

Several researches have been exploring this field, trying to evaluate sources of information in Q&AC. Such as Bouguessa et al. [START_REF] Smets | The Transferable Belief Model[END_REF] who proposed a model to identify authoritative users based on the number of best answers provided by them. A best answer is selected either by the asker or by other users via a voting procedure. In [START_REF] Cambria | Do not feel the trolls[END_REF], the author focused on the selection of questions a user would choose for answering. Based on these studies, experts prefer answering questions where they have a higher chance of making a valuable contribution. Recently in [START_REF] Ortega | Propagation of trust and distrust for the detection of trolls in a social network[END_REF], the authors proposed a framework for evaluating both the reliability and the expertise of an information provider. Considering some cognitive and behavioral criteria of the users, they were able to establish a trust system. Using a response matrix summarizing the interactions between peers of persons, each one is capable of estimating and providing an opinion. Using the subjective logic to aggregate these evaluations, they provided later a global reliability and expertise value for each user within Q&AC.

C. Uncertainty within Q&AC

When dealing with information provided by humans, we are facing several levels of uncertainty. Gjergji et al. proposed three levels for Q&AC [START_REF] Pelechrinisn | Automatic Evaluation of Information Provider Reliability and Expertise[END_REF], the first one is related to the extraction and integration of uncertainty, the second deals with information sources uncertainty and finally the inherent knowledge related to the information itself. In our case, we are more interested in the evaluation of the sources and the part of uncertainty related to them. The main issue in these communities is that we are facing users that we do not always have an apriori knowledge about them. We ignore every thing about the sources' credibility, reliability, relevance, objectivity and expertise. In this context, we will exploit all the mathematical background and large panel of sepcificities provided by the theory of belief functions to help us considering this problem in an uncertain point of view.

III. THEORY OF BELIEF FUNCTIONS

This section recalls the necessary background related to the belief function theory It has been developed by Dempster in his work on upper and lower probabilities [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF]. Based on that, he was able to represent more precisely the observed data.

A belief function must take into consideration all the possible events on which a source can describe a belief. Based on that, we can define the frame of discernment.

A. Frame of discernment

It is a finite set of disjoint elements noted Ω where Ω = {ω 1 , ..., ω n }. This theory allows us to affect a mass on a set of hypotheses not only a singleton like in the probabilistic theory. Thus, we are able to represent ignorance, imprecision...

B. Basic belief assignment (bba)

A bba is defined on the set of all subsets of Ω, named power set and noted 2 Ω . It affects a real value from [0, 1] to every subset of 2 Ω reflecting sources amount of belief on this subset. A bba m verifies:

X⊆Ω m(X) = 1. (1)
We consider any positive elementary mass m(X) > 0 as a focal element such that X belongs to 2 Ω .

C. Combination rules

Many combination rules have been proposed taking in consideration the nature of the sources.

1) Dempster's combination rule: The first one was proposed by Dempster in 1967 [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF] which is a conjunctive normalized combination rule also called the orthogonal sum. Given two mass functions m 1 and m 2 , for all X ∈ 2 Ω , X = ∅, the Dempster's rule is defined by:

m D (X) = m 1 ∩ m 2 (X) = 1 1 -k Y1∩Y2=X m 1 (Y 1 )m 2 (Y 2 ) (2)
where

k = Y1∩Y2=∅ m 1 (Y 1 )m 2 (Y 2 )
is the inconsistency of the fusion (or of the combination) can also be called the conflict or global conflict. (1k) is the normalization factor of the combination in a closed world.

2) The conjunctive combination rule: In order to consider the issues of the open world, the conjunctive combination rule was introduced by Smets [START_REF] Gjergji | CoBayes: bayesian knowledge corroboration with assessors of unknown areas of expertise[END_REF]. Considering two mass functions m 1 and m 2 , for all X ∈ 2 Ω m conj is defined by:

m conj (X) = Y1∩Y2=X m 1 (Y 1 )m 2 (Y 2 )
(3)

3) The disjunctive combination rule: First introduced by Dubois and Prade 1986 [START_REF] Mcqueen | Cluster analysis of multivariate data : Efficiency vs. Interpretability of Classifications[END_REF], the induced results of two bbas m 1 and m 2 is defined as follows:

∀X ⊆ Ωm disj (X) = Y1∪Y2=X m 1 (Y 1 )m 2 (Y 2 ) (4)
The disjunctive combination rule can be used when one of the sources is reliable or when we have no knowledge about their reliability.

IV. INCLUSION AS A CONFLICT MEASURE FOR BELIEF

FUNCTIONS

Recently Martin in [START_REF] Martin | About Conflict in the Theory of Belief Functions, 2nd International Conference on Belief Functions[END_REF] used a degree of inclusion as involved in the measurement made in order to determine the conflict during the combination of two belief functions. He presented an index of inclusion having binary values where:

Inc(X 1 , Y 2 ) = 1, ifX 1 ⊆ Y 2 0, otherwise (5)
With X 1 , Y 2 being respectively the focal elements of m 1 and m 2 . This index is then used to measure the degree of inclusion of the two mass functions and defined as:

d inc = 1 |F 1 ||F 2 | X1∈F1 Y2∈F2 Inc(X 1 , Y 2 ) (6)
Where |F 1 | and |F 2 | are the number of focal elements of m 1 and m 2 . He define the degree of inclusion of m 1 and m 2 : σ inc (m 1 , m 2 ) as follows:

σ inc (m 1 , m 2 ) = max(d inc (m 1 , m 2 ), d inc (m 2 , m 1 )) (7)
Where d inc is the degree of inclusion of m 1 in m 2 and inversely. This inclusion is used as a conflict measure for two mass functions, using it like presented:

Conf (m 1 , m 2 ) = (1 -σ inc (m 1 , m 2 )d(m 1 , m 2 )) (8)
where d(m 1 , m 2 ), is the distance of Jousselme [START_REF] Pal | Expert identification in community question answering: exploring question selection bias[END_REF]:

d(m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D(m 1 -m 2 ) ( 9
)
where D is a metric based on the measure of Jaccard:

D(A, B) = 1, if A = B = ∅ |A∩B| |A∪B| , ∀A, B ∈ 2 Ω (10)

V. TROLLS IDENTIFICATION BASED IN A CONFLICT

MEASURE

Based on the assumption that consists of the trolls' integration in the successful discussion threads, we propose a new method for detecting malicious people in online communities forums. This approach is defined within the framework of belief functions. Indeed, it is based on a conflict measure of this theory applied between the different messages of the thread. We can summarize our proposed method in three major steps that will be discussed in depth in the following.

A. Users' messages

Hardarker proposed primary characteristics of a troll [2] (Aggression, Deception, Disruption, Success). In 2014, Buckels et al. [START_REF] Bouguessa | Identifying authoritative actors in question answering forums: the case of Yahoo! answers[END_REF] specified the behavioral characteristics of a troll. They described them as persons having sadism, psychopathy and machiavilism. To them, trolling is a "deceptive, destructive or disruptive manner in social media". In the context of this work, to distinguish between the troll and the other users, we tried to manually extracted the characteristics of their responses from the answers and comments in different forums. Based on these characteristics, the content of the messages can be: Off-topic, senseless or controversy. Using these characteristics, we have defined the frame of discernment that can characterize a message in a forum:

Ω msg = {Of f -topic, Senseless, 1, . . . , N } (11)
• Senseless: how much the response is empty of meaning?

• Off-topic: How irrelative the answer can be?

• [1..N ] : number of topics where, [1..N ]\i with i being the relevant topic, and [1.

.N ]\i are the controversy topics posted by a troll. During this step, we assume that a method of analysis expresses a piece of evidence concerning the nature of each message. This method aims to analyze the messages relative to the posted question or topic.

B. Users' conflict

Detecting irrelevant messages does not only means that this user is a troll. Thus, it is not only the content of the messages that can characterize the trolls. We can find a victim user that responds to a message posted by a troll. Besides, the subject of the discussion can change gradually. In fact, to distinguish between trolls and other users in a community, we need to quantify how a given user is in conflict with the rest of all the other users. Thus, we will base our approach on measuring the conflict between the messages of each person posting answers. The list of notations is shown in table I. Using the inclusion Contains the sum of conflict of each user as a conflict measure for belief functions, for each user U i we will measure:

• Conf msg /U : measures the conflict between the k th message posted by U i and the messages that were posted before it by each other users U j .

Conf msg /U (m k (U i ), m(U j )) = 1 N P j N Pj s=1 Conf (m k (U i ), m s (U j )), (i = j) (12)
• Conf msg : measures the conflict between the k th message posted by U i and the all messages that were posted before it by all the other users U based on a weighted mean. This measure takes into account the number of messages posted by every user in order to determine the level of conflict especially between a troll and an expert.

Conf msg (m k (U i ), m(U )) = N j=1 N P j N P Conf msg /U (m k (U i ), m(U j )) (13)
• Conf user : measures the global conflict of the user

U i Conf user = 1 N i Ni k=1 Conf msg (m k (U i ), m(U )) (14)
The value of the total conflict of a user can be risen when this user launches into an interminable debate with a troll. In this case, this victim user becomes in his turn a troll. Thus, the managers have to control the behavior of the users in many discussion threads.

C. Users' clustering

The last step consists on the classification of the users according to their conflict results into two groups. Therefore, to make decision we base our approach on an unsupervised classification method using the k means algorithm.

It was introduced by McQueen [START_REF] Zadeh | Fuzzy sets[END_REF] and implemented in its current forms by Forgy [START_REF] Dubois | Possibility Theory[END_REF]. The Kmeans algorithm aims to construct from the objects of the training set K partitions (clusters) concentrated and isolated from each other. In our case, we will devise the users into two partitions: K= 2. Since the value of the troll 's conflict is bigger than the conflict of the other users:

-Trolls belong to the group having the biggest value of center.

-The other users belong to the group having the least value of center.

VI. EXPERIMENTATION

To illustrate the comportment of our proposed method, we have tested it in different simulated data. In this section, we will present two different examples.

A. Example 1

As we presented our method, it has three main steps. Indeed, we will present the results of each step:

1) Users' messages: Our assumption consists on the integration of the trolls in the successful discussion threads. From this point of view, we simulate the data of analyze of messages as depicted in Figure 1. In fact, in this example we will try to detect a troll in a group of 4 users. In this scenario, the discussion thread contains 16 messages posted by different users and among whom three messages are published by a troll. In this example, our frame of discernment is composed by 4 elements: Relevant=X 1 , off-topic=X 2 , senseless=X 3 , controversy-topic=X 4 . As shown in Figure 2 each row presents: the owner of the message, the order of the message in the discussion thread and the mass function of this message (as mention in section III. B each bba must be equal to 1).

In this example, the first message of the troll (U 4 ) is controversy: m(X 4 ) = 0.9210. His second message is empty of meaning: m(X 3 ) = 0.9716. His third message is controversy: m(X 4 ) = 0.8387.

2) Users conflict: Based on the method of the inclusion and applying our algorithm, we will present the total conflict of each user of our example in Table II: U 4 has the biggest value of conflict. The total conflicts of users U 1 and U 2 is small relative to the total conflict of U 4 despite the fact that they responded to the first message of the troll by posting each one a controversy message. this result can be explained by the answers provided by these two users who have published relevant messages. U 3 has a small value of conflict, he published three relevant messages where in his first message m(X 1 ) = 0.9732, in his second message m(X 1 ) = 0.7782, and his third message m(X 1 ) = 0.9632. 3) Users clustering: Applying the K-means algorithm to the different values of total conflict of all users we obtained two clusters.

-Trolls= {U 4 } -Other users= {U 1 , U 2 , U 3 } Our proposal provides us a correct classification of the users. This result shows the feasibility of our proposed method.

B. Example 2

For this simulation we will assume that we are dealing with 8 users, among them two trolls. The discussion thread contains 31 messages. The result of the total conflict of each user expressed in equation 14 is illustrated in figure 2.

The first troll U 4 published 2 controversy messages and the second troll U 8 published 3 messages: The two first ones are off-topic, and the last one is controversy.

-U 1 posted 3 relevant messages and 2 controversy messages to respond to the first troll.

-U 2 posted 7 relevant messages and 2 controversy messages to respond to the first troll. -U 3 posted 4 relevant messages and one off-topic message to respond to the second troll.

-U 5 posted one relevant message. -U 6 published 3 relevant messages. -U 7 published 2 relevant messages.

The total conflict of the troll U 4 is bigger relatively to the other troll U 8 because he published his posts after a big number of reliable messages provided by the other users. So, this situation created a higher value of a conflict. Applying the Kmeans algorithm our method provides us a correct classification:

-Trolls= {U 4 , U 8 } -Other users= {U 1 , U 2 , U 3 , U 5 , U 6 , U 7 } The users U 1 , U 2 and U 3 are not classified among the trolls in spite of their posts that can be categorized as trolls' messages. This result is explained by the fact that they have other relevant messages. We proposed in this paper a new method for detecting 'Trolls' in Q&AC. Relying on this approach managers, can control the behavior of the users in many discussion threads in order to notify them to stop trolling. Our work is defined within an uncertain framework. It is based on a conflict measure in the belief function theory applied between the messages of the different users of the thread. First of all, this method aims to analyze the messages relative to the posted question or topic. But detecting irrelevant message is not enough to judge if this user is a troll or not. Thus, not only the content of the messages that can characterize the trolls but also their behaviors. Next, using the results of this analysis we measured the conflict between the different users. Finally, after calculating the conflict of each user we applied the kmeans method in order to distinguish trolls from the other users. Indeed, we have classified the users according to their conflict results into two clusters. This method was tested in different simulated data to check its feasibility. Since our proposed method for detecting malicious users dealt only with one discussion thread, we aim to extend this approach to detect trolls inside the community.

Fig. 1 .

 1 Fig. 1. Simulation Results

Fig. 2 .

 2 Fig. 2. The steps of detecting trolls

TABLE II TOTAL

 II CONFLICT OF EACH USER

		U 1	U 2	U 3	U 4
	Confuser	0.0610	0.0639	0.0489	0.2030


Detais A • Dupuy
email: detais@math.univ-toulouse.fr

Detais Amélie

Jean-François Dupuy
email: dupuy@math.univ-toulouse.fr

Maximum likelihood estimation in a partially observed stratified regression model with censored data

Keywords: Censored data, Maximum likelihood estimation, Missing data, Stratified intensity model

The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained.

Introduction

This paper considers the problem of estimation in the stratified proportional intensity regression model for survival data, when the stratum information is missing for some sample individuals.

The stratified proportional intensity model (see [START_REF] Andersen | Statistical models based on counting processes[END_REF] or [START_REF] Martinussen | Dynamic regression models for survival data[END_REF] for example) generalizes the usual [START_REF] Cox | Regression models and life-tables (with discussion)[END_REF] proportional intensity regression model for survival data, by allowing different groups -the strata-of the population under study to have distinct baseline intensity functions. More precisely, in the stratified model, the strata divide the sample individuals into K disjoint groups, each having a distinct baseline intensity function λ k but a common value for the regression parameter.

The intensity function for the failure time T 0 of an individual in stratum k thus takes the form λ k (t) exp(β X),

(1) where X is a p-vector of covariates, β is a p-vector of unknown regression parameters of interest, and {λ k (t) : t ≥ 0, k = 1, . . . , K} are K unknown baseline intensity functions.

A consistent and asymptotically normal estimator of β can be obtained by maximizing the partial likelihood function [START_REF] Cox | Partial likelihood[END_REF]. The partial likelihood for the stratified model ( 1) is the product over strata of the within-stratum partial likelihoods (we refer to [START_REF] Andersen | Statistical models based on counting processes[END_REF] for a detailed treatment of maximum partial likelihood estimation in model ( 1)). In some applications, it can also be desirable to estimate the cumulative baseline intensity functions Λ k = λ k . The so-called [START_REF] Breslow | Discussion of the paper by D. R. Cox[END_REF] estimators are commonly used for that purpose (see chapter 7 of [START_REF] Andersen | Statistical models based on counting processes[END_REF] for further details on the Breslow estimator and its asymptotic properties).

One major motivation for using the stratified model is that it allows to accomodate in the analysis a predictive categorical covariate whose effect on the intensity is not proportional. To this end, the individuals under study are stratified with respect to the categories of this covariate. In many applications however, this covariate may be missing for some sample individuals (for example, histological stage determination may require biopsy and due to expensiveness, may not be performed on all the study subjects). In this case, the usual statistical inference for model (1), based on the product of withinstratum partial likelihoods, can not be directly applied.

In this work, we consider the problem of estimating β and the Λ k , k = 1, . . . , K in model ( 1), when the covariate defining the stratum is missing for some (but not all) individuals. Equivalently said, we consider the problem of estimating model (1) when the stratum information is only partially available.

The problem of estimation in the (unstratified) Cox model λ(t) exp(β X) with missing covariate X has been the subject of intense research over the past decade: see for example [START_REF] Lin | Cox regression with incomplete covariate measurements[END_REF], [START_REF] Paik | Multiple imputation for the Cox proportional hazards model with missing covariates[END_REF], [START_REF] Paik | On using the Cox proportional hazards model with missing covariates[END_REF], [START_REF] Chen | Proportional hazards regression with missing covariates[END_REF], [START_REF] Martinussen | Cox regression with incomplete covariate measurements using the EM-algorithm[END_REF], [START_REF] Pons | Estimation in the Cox model with missing covariate data[END_REF], and the references therein. But to the best of our knowledge and despite its practical relevance, the problem of statistical inference in model (1) with partially available stratum information has not been yet extensively investigated. Recently, [START_REF] Dupuy | A study of regression calibration in a partially observed stratified Cox model[END_REF] studied the asymptotic properties of a regression calibration estimator of β in this setting (regression calibration is a general method for handling missing data in regression models, see [START_REF] Carroll | Measurement error in nonlinear models[END_REF] for example). The authors proved that this estimator is asymptotically biased, although nevertheless asymptotically normal. No estimators of the cumulative baseline intensity functions were provided.

In this work, we aim at providing an estimator of β that is both consistent and asymptotically normal. Moreover, although the cumulative intensity functions Λ k are usually not the primary parameters of interest, we also aim at providing consistent and asymptotically normal estimators of the values Λ k (t), k = 1, . . . , K.

The regression calibration inferential procedure investigated by [START_REF] Dupuy | A study of regression calibration in a partially observed stratified Cox model[END_REF] is essentially based on a modified version of the partial likelihood for model (1). In this paper, we propose an alternative method which may be viewed as a fully maximum likelihood approach. Besides assuming that the failure intensity function for an individual in stratum k is given by model (1), we assume that the probability of being in stratum k conditionally on a set of observed covariates W (which may include some components of X) is of the logistic form, depending on some unknown finite-dimensional parameter γ.

A full likelihood for the collected parameter θ = (β, γ, Λ k ; k = 1, . . . , K) is constructed from a sample of incompletely observed data. Based on this, we propose to estimate the finite and infinite-dimensional components of θ by using the nonparametric maximum likelihood (NPML) estimation method. We then provide asymptotic results for these estimators, including consistency, asymptotic normality, semiparametric efficiency of the NPML estimator of β, and consistent variance estimation.

Our proofs use some techniques developed by [START_REF] Murphy | Consistency in a proportional hazards model incorporating a random effect[END_REF][START_REF] Murphy | Asymptotic theory for the frailty model[END_REF] and [START_REF] Parner | Asymptotic theory for the correlated gamma-frailty model[END_REF] to establish the asymptotic theory for the frailty model.

The paper is organized as follows. In Section 2, we describe in greater detail the data structure and the model assumptions. In Section 3, we describe the NPML estimation method for our setting and we establish existence of the NPML estimator of θ. Section 4 establishes the consistency and asymptotic normality of the proposed estimator. Consistent variance estimators are also obtained for both the finite-dimensional parameter estimators and the nonparametric cumulative baseline intensity estimators. We give some concluding remarks in Section 5. Proofs are given in Appendix.

Data structure and model assumptions

We describe the notations and model assumptions that will be used throughout the paper.

All the random variables are defined on a probability space (Ω, C, P). Let T 0 be a random failure time whose distribution depends on a vector of covariates X ∈ R p and on a stratum indicator S ∈ K = {1, . . . , K}. We assume that conditionally on X and S = k (k ∈ K), the intensity function of T 0 is given by model (1). We suppose that T 0 may be right-censored by a positive random variable C and that the analysis is restricted to the time interval [0, τ ], where τ < ∞ denotes the end of the study. Thus we actually observe the potentially censored duration T = min{T 0 , min(C, τ )} and a censoring indicator

∆ = 1{T 0 ≤ min(C, τ )}. If t ∈ [0, τ ],
we denote by N (t) = 1{T ≤ t}∆ and Y (t) = 1{T ≥ t} the failure counting and at-risk processes respectively.

Let W ∈ R m be a vector of surrogate covariates for S (W and X may share some common components). That is, W brings a partial information about S when S is missing, and it adds no information when S is observed so that the distribution of T 0 conditionally on X, S, and W does not involve the components of W that are not in X. We assume that the conditional probability that an individual belongs to the k-th stratum given his covariate vector W follows a multinomial logistic model:

P(S = k|W ) = exp(γ k W ) K j=1 exp(γ j W )
,

where γ k ∈ R m (k ∈ K).
Finally, we let R denote the indicator variable which is 1 if S is observed and 0 otherwise. Then, the data consist of n i.i.d. replicates

O i = (T i , ∆ i , X i , W i , R i , R i S i ), i = 1, . . . , n, of O = (T, ∆, X, W, R, RS). The data available for the i-th individual are therefore (T i , ∆ i , X i , W i , S i ) if R i = 1 and (T i , ∆ i , X i , W i ) if R i = 0.
In the sequel, we set γ K = 0 for model identifiability purpose and we note γ = (γ 1 , . . . , γ K-1 ) ∈ (R m ) K-1 ≡ R q . We also note π k,γ (W ) = P(S = k|W ), k ∈ K. Now, let θ = (β, γ, Λ k ; k ∈ K) be the collected parameter and θ 0 = (β 0 , γ 0 , Λ k,0 ; k ∈ K) denote the true parameter value. Under the true value θ 0 , the expectation of random variables will be denoted P θ0 . P n will denote the empirical probability measure. In the sequel, the stochastic convergences will be in terms of outer measure.

We now make the following additional assumptions:

(a) The censoring time C is independent of T 0 given (S, X, W ), of S given (X, W ), and is non-informative. With probability 1, P(C ≥ T 0 ≥ τ |S, X, W ) > c 0 for some positive constant c 0 . (b) The parameter values β 0 and γ 0 lie in the interior of known compact sets B ⊂ R p and G ⊂ R q respectively. For every k ∈ K, the cumulative baseline intensity function Λ k,0 is a strictly increasing function on [0, τ ] with Λ k,0 (0) = 0 and Λ k,0 (τ ) < ∞. Moreover, for every k ∈ K, Λ k,0 is continuously differentiable in [0, τ ], with λ k,0 (t) = ∂Λ k,0 (t)/∂t. Let A denote the set of functions satisfying these properties. (c) The covariate vectors X and W are bounded (i.e. X < c 1 and W < c 1 , for some finite positive constant c 1 , where • denotes the Euclidean norm). Moreover, the covariance matrices of X and W are positive definite. Let c 2 = min β∈B, X <c1 e β X and c 3 = max β∈B, X <c1 e β X . (d) There is a constant c 4 > 0 such that for every k ∈ K, P θ0 [1{S = k}Y (τ )R] > c 4 , and the sample size n is large enough to ensure that

n i=1 1{S i = k}Y i (τ )R i > 0 for every k ∈ K.
(e) With probability 1, there exists a positive constant c 5 such that for every k ∈ K, P θ0 [∆R1{S = k}|T, X, W ] > c 5 . (f) R is independent of S given W , of (T, ∆) given (X, S). The distribution of S conditionally on X and W does not involve the components of X that are not in W . The distributions of R and of the covariate vectors X and W do not depend on the parameter θ.

Remark. Conditions (b), (c), (d), and (e) are used for identifiability of the parameters and consistency of the proposed estimators. Condition (d) essentially requires that for every stratum k, some subjects are known to belong to k and are still at risk when the study ends. The first assumption in condition (f) states that S is missing at random, which is a fairly general missing data situation (we refer to chapters 6 and 7 in [START_REF] Tsiatis | Semiparametric theory and missing data[END_REF] for a recent exposition of missing data mechanisms).

Remark. We are now in position to describe our proposed approach to the problem of estimation in model (1) from a sample of incomplete data O i , i = 1, . . . , n.

Let S denote the set of subjects with unknown stratum in this sample. The regression calibration method investigated by [START_REF] Dupuy | A study of regression calibration in a partially observed stratified Cox model[END_REF] essentially allocates every subject of S to each of the strata, and estimates β 0 by maximizing a modified version of the partial likelihood for the stratified model, where the contribution of any individual i in S to the within-k-thstratum partial likelihood is weighted by an estimate of π k,γ (W i ) (for every k ∈ K). The asymptotic bias of the resulting estimator arises from the failure of this method to fully exploit the information carried by (T i , ∆ i , X i , W i ) on the unobserved stratum indicator S i .

Therefore in this paper, we rather suggest to weight each subject i in S by an estimate of the conditional probability that subject i belongs to the k-th stratum given the whole observed data (T i , ∆ i , X i , W i ). This suggestion raises two main problems, as is described below.

Remark. First, we should note that the suggested alternative weights depend on the unknown baseline intensity functions. Therefore, the modified partial likelihood approach considered by [START_REF] Dupuy | A study of regression calibration in a partially observed stratified Cox model[END_REF] can not be used to derive an estimator for β 0 . Next, the statistics to be involved in the score function for β will depend on the conditional weights and thus, this score will not be expressible as a stochastic integral of some predictable process, as is often the case in models for failure time data. This, in turn, will prevent us from using the counting process martingale theory usually associated with the theoretical developments in failure time models.

To overcome the first problem, we define our estimators from a full likelihood for the whole parameter, that is, for both the finite-dimensional -β (and γ)-and infinite-dimensional -Λ k , k ∈ K-components of θ. Empirical process theory (van der Vaart and Wellner 1996) is used to establish asymptotics for the proposed estimators.

Maximum likelihood estimation

In the sequel, we assume that there are no ties among the observed death times (this hypothesis is made to simplify notations, but the results below can be adapted to accomodate ties). The likelihood function for observed data O i , i = 1, . . . , n is given by

L n (θ) = n i=1 K k=1 λ k (T i ) ∆i exp ∆ i β X i -e β Xi Λ k (T i ) π k,γ (W i ) 1{Si=k} Ri × K k=1 λ k (T i ) ∆i exp ∆ i β X i -e β Xi Λ k (T i ) π k,γ (W i ) 1-Ri (2)
It would seem natural to derive a maximum likelihood estimator of θ 0 by maximizing the likelihood (2). However, the maximum of this function over the parameter space Θ = B × G × A ⊗K does not exist. To see this, consider functions Λ k with fixed values at the T i , and let (∂Λ

k (t)/∂t)| t=Ti = λ k (T i ) go to infinity for some T i with ∆ i R i 1{S i = k} = 1 or ∆ i (1 -R i ) = 1.
To overcome this problem, we introduce a modified maximization space for (2), by relaxing each Λ k (•) to be an increasing right-continuous step-function on [0, τ ], with jumps at the T i 's such that

∆ i R i 1{S i = k} = 1 or ∆ i (1 -R i ) = 1.
Estimators of (β 0 , γ 0 , Λ k,0 ; k ∈ K) will thus be derived by maximizing a modified version of (2), obtained by replacing λ k (T i ) in ( 2) with the jump size

Λ k {T i } of Λ k at T i .
If they exist, these estimators will be referred to as nonparametric maximum likelihood estimators -NPMLEs -(we refer to [START_REF] Zeng | Maximum likelihood estimation in semiparametric regression models with censored data[END_REF] for a review of the general principle of NPML estimation, with application to various semiparametric regression models for censored data. See also the numerous references therein). In our setting, existence of such estimators is ensured by the following theorem (proof is given in Appendix):

Theorem 1 Under conditions (a)-(f ), the NPMLE θ n = ( β n , γ n , Λ k,n ; k ∈ K) of θ 0 exists and is achieved.
The problem of maximizing L n over the approximating space described above reduces to a finite dimensional problem, and the expectation-maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] can be used to calculate the NPMLEs. For 1 ≤ i ≤ n and k ∈ K, let w i (k, θ) be the conditional probability that the i-th individual belongs to the k-th stratum given (T i , ∆ i , X i , W i ) and the parameter value θ, and let Q(O i , k, θ) denote the conditional expectation of 1{S i = k} given O i and the parameter value θ. Then Q(O i , k, θ) has the form

Q(O i , k, θ) = R i 1{S i = k} + (1 -R i )w i (k, θ).
In the M-step of the EM-algorithm, we solve the complete-data score equation conditional on the observed data. In particular, the following expression for the NPMLE of Λ k (•) can be obtained by: (a) taking the derivative with respect to the jump sizes of Λ k (•), of the conditional expectation of the complete-data log-likelihood given the observed data and the NPML estimator, (b) setting this derivative equal to 0:

Lemma 1 The NPMLE θ n satisfies the following equation for every k ∈ K:

Λ k,n (t) = t 0 n i=1 Q(O i , k, θ n ) n j=1 Q(O j , k, θ n ) exp( β n X j )Y j (s) dN i (s), 0 ≤ t ≤ τ.
The details of the calculations are omitted (note how the suggested weights w i (k, θ) naturally arise from the M-step of the EM algorithm). We refer the interested reader to [START_REF] Zeng | Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time[END_REF] and [START_REF] Sugimoto | Properties of estimators of baseline hazard functions in a semiparametric cure model[END_REF], who recently described EM algorithms for computing NPMLEs in various other semiparametric models with censored data.

In the sequel, we shall denote the conditional expectation of the completedata log-likelihood given the observed data and the NPML estimator by

E n [ l n (θ)].

Asymptotic properties

This section states the asymptotic properties of the proposed estimators. We first obtain the following theorem, which states the strong consistency of the proposed NPMLE. The proof is given in Appendix.

Theorem 2 Under conditions (a)-(f ), β n -β 0 , γ n -γ 0 , and sup t∈[0,τ ] | Λ k,n (t)- Λ k,0 (t)| (for every k ∈ K) converge to 0 almost surely as n tends to infinity.
To derive the asymptotic normality of the proposed estimators, we adapt the function analytic approach developed by [START_REF] Murphy | Asymptotic theory for the frailty model[END_REF] for the frailty model (see also [START_REF] Chang | An asymptotic theory for the nonparametric maximum likelihood estimator in the Cox gene model[END_REF], [START_REF] Kosorok | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF], and [START_REF] Lu | Maximum likelihood estimation in the proportional hazards cure model[END_REF], for recent examples of this approach in various other models).

Instead of calculating score equations by differentiating E n [ l n (θ)] with respect to β, γ, and the jump sizes of Λ k (•), we consider one-dimensional submodels θ n,η passing through θ n and we differentiate with respect to η. Precisely, we consider submodels of the form

η → θ n,η = β n + ηh β , γ n + ηh γ , • 0 {1 + ηh Λ k (s)}d Λ k,n (s); k ∈ K ,
where h β and h γ = (h γ1 , . . . , h γ K-1 ) are p-and q-dimensional vectors respectively (h γj ∈ R m , j = 1, . . . , K -1), and the

h Λ k (k ∈ K) are functions on [0, τ ]. Let h = (h β , h γ , h Λ k ; k ∈ K).
To obtain the score equations, we differentiate E n [ l n ( θ n,η )] with respect to η and we evaluate at η = 0.

θ n maximizes E n [ l n (θ)] and therefore satisfies (∂E n [ l n ( θ n,η )]/∂η) | η=0 = 0 for every h, which leads to the score equation S n ( θ n )(h) = 0 where S n ( θ n )(h) takes the form S n ( θ n )(h) = P n h β S β ( θ n ) + h γ S γ ( θ n ) + K k=1 S Λ k ( θ n )(h Λ k ) , (3)
where

S β (θ) = ∆X - K k=1 Q(O, k, θ)X exp(β X)Λ k (T ), S γ (θ) = (S γ1 (θ) , . . . , S γ K-1 (θ) ) with S γ k (θ) = W [Q(O, k, θ) -π k,γ (W )] , S Λ k (θ)(h Λ k ) = Q(O, k, θ) h Λ k (T )∆ -exp(β X) T 0 h Λ k (s) dΛ k (s) .
We take the space of elements h = (h β , h γ , h Λ k ; k ∈ K) to be

H = {(h β , h γ , h Λ k ; k ∈ K) : h β ∈ R p , h β < ∞; h γ ∈ R q , h γ < ∞; h Λ k is a function defined on [0, τ ], h Λ k v < ∞, k ∈ K},
where h Λ k v denotes the total variation of h Λ k on [0, τ ]. We further take the functions h Λ k to be continuous from the right at 0. Define

θ(h) = h β β + h γ γ + K k=1 τ 0 h Λ k (s) dΛ k (s)
, where h ∈ H. From this, the parameter θ can be considered as a linear functional on H, and the parameter space Θ can be viewed as a subset of the space l ∞ (H) of bounded real-valued functions on H, which we provide with the uniform norm. Moreover, the score operator S n appears to be a random map from Θ to the space l ∞ (H). Note that appropriate choices of h allow to extract all components of the original parameter θ. For example, letting h γ = 0, h Λ k (•) = 0 for every k ∈ K, and h β be the p-dimensional vector with a one at the i-th location and zeros elsewhere yields the i-th component of β.

Letting h β = 0, h γ = 0, h Λ k (•) = 0 for every k ∈ K except h Λj (s) = 1{s ≤ t} (for some t ∈ (0, τ )) yields Λ j (t).
We need some further notations to state the asymptotic normality of the NPMLE of β 0 . Let us first define the linear operator σ

= (σ β , σ γ , σ Λ k ; k ∈ K) : H → H by σ β (h) = P θ0 [2X∆ψ(O, θ 0 ) K k=1 Q(O, k, θ 0 )h Λ k (T )] +P θ0 [ψ(O, θ 0 )X {ψ(O, θ 0 )X h β + S γ (θ 0 ) h γ }] , σ γ (h) = P θ0 [2S γ (θ 0 )∆ K k=1 Q(O, k, θ 0 )h Λ k (T )] + P θ0 [S γ (θ 0 )S γ (θ 0 ) ] h γ +P θ0 [ψ(O, θ 0 )S γ (θ 0 )X ] h β , σ Λ k (h)(u) = h Λ k (u)P θ0 [Q(O, k, θ 0 )φ(u, O, k, θ 0 )] +P θ0 [2φ(u, O, k, θ 0 ) j>k Q(O, j, θ 0 ){h Λj (u) -e β 0 X u 0 h Λj dΛ j,0 -∆h Λj (T ) + e β 0 X T 0 h Λj dΛ j,0 }] -h β P θ0 [2Xψ(O, θ 0 )Q(O, k, θ 0 )e β 0 X Y (u)] -h γ P θ0 [2S γ (θ 0 )Q(O, k, θ 0 )e β 0 X Y (u)],
where

φ(u, O, k, θ 0 ) = Y (u)Q(O, k, θ 0 )e β 0 X and ψ(O, θ 0 ) = ∆ - K k=1 Q(O, k, θ 0 )e β 0 X Λ k,0 (T ).
This operator is continuously invertible (Lemma 3 in Appendix). We shall denote its inverse by σ

-1 = (σ -1 β , σ -1 γ , σ -1 Λ k ; k ∈ K).
Next, for every r ∈ N\{0}, the r-dimensional column vector having all its components equal to 0 will be noted by 0 r (or by 0 when no confusion may occur

). Let h = (h β , h γ , h Λ k ; k ∈ K) ∈ H. If h γ = 0 and h Λ k is identically equal to 0 for every k ∈ K, we note h = (h β , 0, 0; k ∈ K). Let σ -1 β : R p → R p be the linear map defined by σ -1 β (u) = σ -1 β ((u, 0, 0; k ∈ K)), for u ∈ R p . Let {e 1 , . . . , e p } be the canonical basis of R p .
Then the following result holds, its proof is given in Appendix.

Theorem 3 Under conditions (a)-(f ), √ n( β n -β 0 ) has an asymptotic normal distribution N (0, Σ β ), where

Σ β = ( σ -1 β (e 1 ), . . . , σ -1 β (e p ))
is the efficient variance in estimating β 0 .

Remark. Although γ 0 and the cumulative baseline intensity functions Λ k,0 (k ∈ K) are not the primary parameters of interest, we may also state an asymptotic normality result for their NMPLEs. This requires some further notations.

Define σ -1 γ : R q → R q by σ -1 γ (u) = σ -1 γ ((0, u, 0; k ∈ K)), let {f 1 , . . . , f q } be the canonical basis of R q , and define Σ γ = ( σ -1 γ (f 1 ), . . . , σ -1 γ (f q )). Finally, let h (j,t) = (h β , h γ , h Λ k ; k ∈ K) be such that h β = 0, h γ = 0, h Λj (•) = 1{• ≤ t} for some t ∈ (0, τ ) and j ∈ K, and h Λ k = 0 for every k ∈ K, k = j. Then the following holds (a brief sketch of the proof is given in Appendix):

Theorem 4 Assume that conditions (a)-(f ) hold. Then √ n( γ n -γ 0 ) has an asymptotic normal distribution N (0, Σ γ ). Furthermore, for any t ∈ (0, τ ) and j ∈ K, √ n( Λ j,n (t) -Λ j,0 (t)) is asymptotically distributed as a N (0, v 2 j (t)), where

v 2 j (t) = t 0 σ -1 Λj (h (j,t) )(u) dΛ j,0 (u).
We now turn to the issue of estimating the asymptotic variances of the estimators β n , γ n , and Λ j,n (t) (t ∈ (0, τ ), j ∈ K). It turns out that the asymptotic variances Σ β , Σ γ , and v 2 j (t) are not expressible in explicit forms, since the inverse σ -1 has no closed form. However, this is not a problem if we can provide consistent estimators for them. Such estimators are defined below.

For i = 1, . . . , n, let X ir denote the r-th (r = 1, . . . , p) component of X i , S γ,i (θ) be defined as in (3) with O and W replaced by O i and W i respectively, and S γ,i,s (θ) be the s-th (s = 1, . . . , q) component of S γ,i (θ). Using these notations, we define the following block matrix

A n =   A ββ A βγ A βΛ A γβ A γγ A γΛ A Λβ A Λγ A ΛΛ   (4)
where the sub-matrices A ββ , A γγ , A βγ , and A γβ are defined as follows by their (r, s)-th component:

A ββ rs = 1 n n i=1 {ψ(O i , θ n )} 2 X ir X is , r, s = 1, . . . , p, A γγ rs = 1 n n i=1 S γ,i,r ( θ n )S γ,i,s ( θ n ), r, s = 1, . . . , q, A βγ rs = 1 n n i=1 ψ(O i , θ n )X ir S γ,i,s ( θ n ), r = 1, . . . , p, s = 1, . . . , q,
A γβ rs = A βγ sr , r = 1, . . . , q, s = 1, . . . , p.

Define the block matrices A βΛ = (A βΛ1 , . . . , A βΛ K ) and A γΛ = (A γΛ1 , . . . , A γΛ K ), where for every k ∈ K, the sub-matrices A βΛ k and A γΛ k are defined by

A βΛ k rs = 2 n X sr ∆ s ψ(O s , θ n )Q(O s , k, θ n ), r = 1, . . . , p, s = 1, . . . , n, A γΛ k rs = 2 n S γ,s,r ( θ n )∆ s Q(O s , k, θ n ), r = 1, . . . , q, s = 1, . . . , n.
Define also the block matrices

A Λβ =    A Λ1β . . . A Λ K β    A Λγ =    A Λ1γ . . . A Λ K γ    A ΛΛ =    A Λ1Λ1 . . . A Λ1Λ K . . . . . . A Λ K Λ1 . . . A Λ K Λ K   
where for every j, k ∈ K,

A Λ k β rs = - 1 n n i=1 2X is ψ(O i , θ n )Q(O i , k, θ n )e β n Xi Y i (T r ), r = 1, . . . , n, s = 1, . . . , p, A Λ k γ rs = - 1 n n i=1 2S γ,i,s ( θ n )Q(O i , k, θ n )e β n Xi Y i (T r ), r = 1, . . . , n, s = 1, . . . , q, A Λ k Λj rs = 1{j = k}1{r = s} 1 n n i=1 Q(O i , k, θ n )φ(T r , O i , k, θ n ) +1{j > k}(1{r = s} 1 n n i=1 2φ(T s , O i , k, θ n )Q(O i , j, θ n ) + 2 n n i=1 φ(T r , O i , k, θ n )Q(O i , j, θ n )e β n Xi ∆Λ j,n (T s ){1{T s ≤ T i } -1{T s ≤ T r }} - 2 n φ(T r , O s , k, θ n )Q(O s , j, θ n )∆ s ), r, s = 1, . . . , n,
and

∆Λ j,n (T s ) is the jump size of Λ j,n at T s that is, ∆Λ j,n (T s ) = Λ j,n (T s ) - Λ j,n (T s -) (j ∈ K, s = 1, . . . , n).
Note that for notational simplicity, the lower (sample size) indice n has been omitted in the notations for the sub-matrices of A n . Now, define

Σ β,n = A ββ -A βγ (A γγ ) -1 A γβ -(A βΛ -A βγ (A γγ ) -1 A γΛ ) ×(A ΛΛ -A Λγ (A γγ ) -1 A γΛ ) -1 (A Λβ -A Λγ (A γγ ) -1 A γβ ) -1 , Σ γ,n = A γγ -A γβ (A ββ ) -1 A βγ -(A γΛ -A γβ (A ββ ) -1 A βΛ ) ×(A ΛΛ -A Λβ (A ββ ) -1 A βΛ ) -1 (A Λγ -A Λβ (A ββ ) -1 A βγ ) -1 ,
and

Σ Λ,n = A ΛΛ -A Λβ (A ββ ) -1 A βΛ -(A Λγ -A Λβ (A ββ ) -1 A βγ ) ×(A γγ -A γβ (A ββ ) -1 A βγ ) -1 (A γΛ -A γβ (A ββ ) -1 A βΛ ) -1 .
Then the following holds:

Theorem 5 Under conditions (a)-(f ), Σ β,n and Σ γ,n converge in probability to Σ β and Σ γ respectively as n tends to ∞. Moreover, for t ∈ (0, τ ) and j ∈ K, let

v 2 j,n (t) = Ξ n (j,t) Σ Λ,n U n (j,t)
, where

Ξ n (j,t) = 0 (j-1)n , ∆Λ j,n (T 1 )1{T 1 ≤ t}, . . . , ∆Λ j,n (T n )1{T n ≤ t}, 0 (K-j)n and U n (j,t) = (0 (j-1)n , 1{T 1 ≤ t}, . . . , 1{T n ≤ t}, 0 (K-j)n ) . Then v 2 j,n (
t) converges in probability to v 2 j (t) as n tends to ∞.

Discussion

In this paper, we have constructed consistent and asymptotically normal estimators for the stratified proportional intensity regression model when the sample stratum information is only partially available. The proposed estimator for the regression parameter of interest in this model has been shown to be semiparametrically efficient. Although computationally more challenging, these estimators improve the ones previously investigated in the literature, such as the regression calibration estimators [START_REF] Dupuy | A study of regression calibration in a partially observed stratified Cox model[END_REF].

We have obtained explicit (and computationally fairly simple) formulas for consistent estimators of the asymptotic variances. These formulas may however require the inversion of potentially large matrices. For a large sample, this inversion may be unstable. An alternative solution relies on numerical differentiation of the profile log-likelihood (see [START_REF] Murphy | Maximum likelihood estimation in the proportional odds model[END_REF] and Chen and Little (1999) for example). Note that in this latter method however, no estimator is available for the asymptotic variance of the cumulative baseline intensity estimator. Some further work is needed to evaluate the numerical performance of the proposed estimators. This is the subject for future research, and requires some extensive simulation work which falls beyond the scope of this paper.

In this paper, a multinomial logistic model [START_REF] Jobson | Applied multivariate data analysis[END_REF]) is used for modeling the conditional stratum probabilities given covariates. This choice was mainly motivated by the fact that this model is commonly used in medical research for modeling the relationship between a categorical response and covariates. The theoretical results developed here can be extended to the case of other link functions. In addition, the covariate X in model ( 1) is assumed to be time independent, for convenience. This assumption can be relaxed to accomodate time varying covariates, provided that appropriate regularity conditions are made.

To show that a maximum of L n exists on B × G × [0, ∞) i n

• , we show that there exists a finite M such that for all θ

M = (β M , γ M , (Λ M k {t (k,j) }) j,k ) ∈ (B × G × [0, ∞) i n • )\(B × G × [0, M ] i n • ), there exists a θ = (β, γ, (Λ k {t (k,j) }) j,k ) ∈ B×G×[0, M ] i n • such that L n (θ) > L n (θ M ).
A proof by contradiction is adopted for that purpose.

Assume that for all M < ∞, there exists

θ M ∈ (B × G × [0, ∞) i n • )\(B × G × [0, M ] i n • ) such that for all θ ∈ B × G × [0, M ] i n • , L n (θ) ≤ L n (θ M ).
It can be seen that L n is bounded above by

K n n i=1   K k=1 {c 3 Λ k {T i }} ∆iRi1{Si=k} exp   -c 2 R i 1{S i = k} i n k j=1 Λ k {t (k,j) }1{t (k,j) ≤ T i }     . If θ M ∈ (B × G × [0, ∞) i n • )\(B × G × [0, M ] i n • )
, then there exists l ∈ K and p ∈ {1, . . . , i n l } such that Λ M l {t (l,p) } > M . By assumption (d), there exists at least one individual with indice i M (i M ∈ {1, . . . , n}) such that 1{S i M = l} = 1, Y i M (τ ) = 1 (and therefore t (l,p) ≤ T i M = τ ), and

R i M = 1. Hence R i M 1{S i M = l} i n l j=1 Λ M l {t (l,j) }1{t (l,j) ≤ T i M } → ∞ as M → ∞.
It follows that the upper bound of L n (θ M ) (and therefore L n (θ M ) itself) can be made as close to 0 as desired by increasing M . This is the desired contradiction.

A.2 Proof of Theorem 2

We adapt the techniques developed by [START_REF] Murphy | Consistency in a proportional hazards model incorporating a random effect[END_REF], in order to prove consistency of our proposed estimator θ n . The proof essentially consists of three steps: (i) for every k ∈ K, we show that the sequence Λ k,n (τ ) is almost surely bounded as n goes to infinity, (ii) we show that every subsequence of n contains a further subsequence along which the NPMLE θ n converges, (iii) we show that the limit of every convergent subsequence of θ n is θ 0 .

Proof of (i). Note first that for all s ∈ [0, τ ] and k ∈ K,

1 n n i=1 Q(O i , k, θ n )e β n Xi Y i (s) ≥ c 2 1 n n i=1 R i 1{S i = k}Y i (τ ). Moreover, Q(O i , k, θ n ) is bounded by 1. It follows that for all k ∈ K, 0 ≤ Λ k,n (τ ) ≤ 1 c 2 τ 0 d Nn (s) 1 n n i=1 R i 1{S i = k}Y i (τ ) = 1 n n i=1 ∆ i c 2 1 n n i=1 R i 1{S i = k}Y i (τ ) , where Nn (s) = n -1 n i=1 N i (s). Next, 1 n n i=1 R i 1{S i = k}Y i (τ ) converges almost surely to P θ0 [R1{S = k}Y (τ )] > c 4 > 0 therefore, for each k ∈ K, as n goes to infinity, Λ k,n (τ ) is bounded above almost surely by 1 c2c4 .
Proof of (ii). If (i) holds, by Helly's theorem (see [START_REF] Loève | Probability theory[END_REF], p179), every subsequence of n has a further subsequence along which Λ 1,n converges weakly to some nondecreasing right-continuous function Λ * 1 , with probability 1. By successive extractions of sub-subsequences, we can further find a subsequence (say n j ) such that Λ k,nj converges weakly to some nondecreasing right-continuous function Λ * k , for every k ∈ K, with probability 1. By the compactness of B ×G, we can further find a subsequence of n j (we shall still denote it by n j for simplicity of notations) such that Λ k,nj converges weakly to Λ * k (for every k ∈ K) and ( β nj , γ nj ) converges to some (β * , γ * ), with probability 1. We now show that the Λ * k 's must be continuous on [0, τ ]. Let ψ be any nonnegative, bounded, continuous function. Then, for any given k ∈ K,

τ 0 ψ(s) dΛ * k (s) = τ 0 ψ(s) d{Λ * k (s) -Λ k,nj (s)} + τ 0 ψ(s) 1 n j nj l=1 Q(O l , k, θ nj )e β n j X l Y l (s) -1 1 n j nj i=1 Q(O i , k, θ nj ) dN i (s) ≤ τ 0 ψ(s) d{Λ * k (s) -Λ k,nj (s)} + τ 0 ψ(s) c 2 n j nj l=1 R l 1{S l = k}Y l (s) -1 d Nnj (s).
By the Helly-Bray Lemma (see [START_REF] Loève | Probability theory[END_REF], p180),

τ 0 ψ(s) d{Λ * k (s)-Λ k,nj (s)} → 0 as j → ∞. Moreover, Nnj (•) and 1 nj nj l=1 R l 1{S l = k}Y l (•) converge almost surely in supremum norm to K k=1 • 0 P θ0 1{S = k}e β 0 X Y (s) dΛ k,0 (s) and P θ0 [R1{S = k}Y (•)]
respectively, where the latter term is bounded away from 0 on s ∈ [0, τ ] by assumption (d). Thus, by applying the extended version of the Helly-Bray Lemma (stated by [START_REF] Korsholm | Likelihood Ratio Test in the Correlated Gamma-Frailty Model[END_REF] for example) to the second term on the right-hand side of the previous inequality, we get that

τ 0 ψ(s) dΛ * k (s) (5) ≤ c 2 τ 0 ψ(s) {P θ0 [R1{S = k}Y (s)]} -1 K k=1 P θ0 [1{S = k}e β 0 X Y (s)]λ k,0 (s) ds. ≤ c 2 c 3 c 4 K k=1 τ 0 ψ(s)λ k,0 (s) ds.
Suppose that Λ * k has discontinuities, and let ψ be close to 0 except at the jump points of Λ * k , where it is allowed to have high and thin peaks. While the right-hand side of inequality ( 5) should be close to 0 (λ k,0 is continuous by assumption (b)), its left-hand side can be made arbitrarily large, yielding a contradiction. Thus Λ * k must be continuous (k ∈ K). A second conclusion, arising from Dini's theorem, is that Λ k,nj uniformly converges to Λ * k (k ∈ K), with probability 1. To summarize: for any given subsequence of n, we have found a further subsequence n j and an element (β * , γ * , Λ * k , k ∈ K) such that β nj -β * , γ nj -γ * , and sup t∈[0,τ ] | Λ k,nj (t) -Λ * k (t)| (for every k ∈ K) converge to 0 almost surely.

Proof of (iii). To prove (iii), we first define random step functions

Λ k,n (t) = t 0 n i=1 Q(O i , k, θ 0 ) n j=1 Q(O j , k, θ 0 ) exp(β 0 X j )Y j (s) dN i (s), 0 ≤ t ≤ τ, k ∈ K,
and we show that for every k ∈ K, Λ k,n almost surely uniformly converges to

Λ k,0 on [0, τ ]. First, note that sup t∈[0,τ ] Λ k,n (t) -P θ0 ∆1{T ≤ t}Q(O, k, θ 0 ) P θ0 1{S = k}e β 0 X Y (s) | s=T ≤ sup t∈[0,τ ] 1 n n i=1 ∆ i 1{T i ≤ t}Q(O i , k, θ 0 ) × 1 P n Q(O, k, θ 0 )e β 0 X Y (s) - 1 P θ0 1{S = k}e β 0 X Y (s) | s=T i + sup t∈[0,τ ] (P n -P θ0 ) ∆1{T ≤ t}Q(O, k, θ 0 ) P θ0 1{S = k}e β 0 X Y (s) | s=T ≤ sup s∈[0,τ ] 1 P n Q(O, k, θ 0 )e β 0 X Y (s) - 1 P θ0 1{S = k}e β 0 X Y (s) + sup t∈[0,τ ] (P n -P θ0 ) ∆1{T ≤ t}Q(O, k, θ 0 ) P θ0 1{S = k}e β 0 X Y (s) | s=T (6)
The class {Y (s) : (Corollary 9.31, Kosorok (2008)), and therefore Glivenko-Cantelli. Moreover,

s ∈ [0, τ ]} is Donsker and Q(O, k, θ 0 )e β 0 X is a bounded measurable function, hence {Q(O, k, θ 0 )e β 0 X Y (s) : s ∈ [0, τ ]} is Donsker
P θ0 [Q(O, k, θ 0 )e β 0 X Y (s)] = P θ0 [P θ0 [1{S = k}|O]e β 0 X Y (s)] = P θ0 [1{S = k}e β 0 X Y (s)]. Thus sup s∈[0,τ ] P n Q(O, k, θ 0 )e β 0 X Y (s) -P θ0 1{S = k}e β 0 X Y (s) converges to 0 a.e. Next, P θ0 [1{S = k}e β 0 X Y (s)] is larger than c 2 .P θ0 [1{S = k}Y (τ )
] and thus, by assumption (d), P θ0 [1{S = k}e β 0 X Y (s)] > 0. It follows that the first term on the right-hand side of inequality (6) converges to 0 a.e.. Similar aguments show that the class

{∆1{T ≤ t}Q(O, k, θ 0 )/P θ0 [1{S = k}e β 0 X Y (s)] | s=T : t ∈ [0, τ ]
} is also a Glivenko-Cantelli class, and therefore Λ k,n almost surely uniformly converges to

P θ0 ∆1{T ≤ t}Q(O, k, θ 0 ) P θ0 1{S = k}e β 0 X Y (s) | s=T . Now, note that Λ k,0 (t) = t 0 P θ 0 [1{S=k} dN (s)] P θ 0 [1{S=k}e β 0 X Y (s)]
, which can be reexpressed as

Λ k,0 (t) = P θ0 [1{S = k}∆1{T ≤ t}] P θ0 1{S = k}e β 0 X Y (s) | s=T = P θ0 ∆1{T ≤ t}Q(O, k, θ 0 ) P θ0 1{S = k}e β 0 X Y (s) | s=T .
Thus Λ k,n almost surely uniformly converges to Λ k,0 on [0, τ ].

Next, using somewhat standard arguments (see [START_REF] Parner | Asymptotic theory for the correlated gamma-frailty model[END_REF] for example), we can show that 0 ≤ n -1 j {log L nj ( θ nj ) -log L nj (θ nj )} converges to the negative Kullback-Leibler information P θ0 [log(L 1 (θ * )/L 1 (θ 0 ))]. Thus, the Kullback-Leibler information must be zero, and it follows that with probability 1, L 1 (θ * ) = L 1 (θ 0 ). The proof of consistency is completed if we show that this equality implies θ * = θ 0 . For that purpose, consider L 1 (θ * ) = L 1 (θ 0 ) under ∆ = 1, R = 1, and 1{S = k} = 1 (for each k ∈ K in turn). Note that this is possible by assumption (e). This yields the following equation for almost all

t ∈ [0, τ ], x < c 1 , w < c 1 : log λ * k (t) λ k,0 (t) + (β * -β 0 ) x -Λ * k (t)e β * x + Λ k,0 (t)e β 0 x + log π * k (w) π k,0 (w) = 0.
This equation is analogous to equation (A.2) in [START_REF] Chen | Proportional hazards regression with missing covariates[END_REF]. The rest of the proof of identifiability thus proceeds along the same lines as the proof of Lemma A.1.1 in [START_REF] Chen | Proportional hazards regression with missing covariates[END_REF], and is omitted. Hence, for any given subsequence of n, we have found a further subsequence n j such that β nj -β 0 , γ nj -γ 0 , and sup t∈[0,τ ] | Λ k,nj (t) -Λ k,0 (t)| (for every k ∈ K) converge to 0 almost surely, which implies that the sequence of NPMLE θ n converges almost surely to θ 0 .

A.3 Proof of Theorem 3

The proof of Theorem 3 uses similar arguments as the proof of Theorem 3 of [START_REF] Fang | Maximum likelihood estimation in a semiparametric logistic/proportional-hazards mixture model[END_REF], so we only highlight the parts that are different. We need a few lemmas before presenting the proof.

Lemma 2 Let h ∈ H. Then the following holds:

P θ0 [S 1 (θ 0 )(h)] = P θ0 [h β S β (θ 0 )+ h γ S γ (θ 0 ) + K k=1 S Λ k (θ 0 )(h Λ k )] = 0.
Proof. From the properties of the conditional expectation, we first note that

P θ0 [S β (θ 0 )] = P θ0 ∆X - K k=1 Q(O, k, θ 0 )X exp(β 0 X)Λ k,0 (T ) = P θ0 ∆X - K k=1 1{S = k}X exp(β 0 X)Λ k,0 (T ) = P θ0 [XM (τ )] ,
where

M (t) = N (t) - t 0 K k=1 1{S = k}e β 0 X Y (u) dΛ k,0 (u)
is the counting process martingale with respect to the filtration F t = σ{N (u), 1{C ≤ u}, X, S, W : 0 ≤ u ≤ t}. X is bounded and F t -measurable, hence it follows that P θ0 [S β (θ 0 )] = 0. Using similar arguments, we can verify that

P θ0 [S Λ k (θ 0 )(h Λ k )] = 0, k ∈ K. Finally, for k = 1, . . . , K -1, P θ0 [S γ k (θ 0 )] = P θ0 [W [Q(O, k, θ 0 ) -π k,γ0 (W )]] = P θ0 [W P θ0 [1{S = k} -π k,γ0 (W )|W ]] = 0.
Combining these results yields that P θ0 [S 1 (θ 0 )(h)] = 0.

We now come to the continuous invertibility of the continuous linear operator σ defined in Section 4.

Lemma 3 The operator σ is continuously invertible.

Proof. Since H is a Banach space, to prove that σ is continuously invertible, it is sufficient to prove that σ is one-to-one and that it can be written as the sum of a bounded linear operator with a bounded inverse and a compact operator (Lemma 25.93 of van der Vaart (1998)).

Define the linear operator

A(h) = (h β , h γ , P θ0 [1{S = k}φ(•, O, k, θ 0 )] h Λ k (•); k ∈ K)
, this is a bounded operator due to the boundedness of X. Moreover, for all u ∈ [0, τ ] and k ∈ K, P θ0 [1{S = k}φ(u, O, k, θ 0 )] ≥ c 2 c 4 > 0 by assumptions (c) and (d). This implies that A is invertible with bounded inverse

A -1 (h) = (h β , h γ , P θ0 [1{S = k}φ(•, O, k, θ 0 )] -1 h Λ k (•); k ∈ K).
The operator σ -A can be shown to be compact by using the same techniques as in [START_REF] Lu | Maximum likelihood estimation in the proportional hazards cure model[END_REF] for example.

To prove that σ is one-to-one, let h ∈ H such that σ(h) = 0. If σ(h) = 0, P θ0 S 1 (θ 0 )(h) 2 = 0, and therefore S 1 (θ 0 )(h) = 0 almost surely. Let j ∈ K.

By assumption (e), for almost every t ∈ [0, τ ], x ≤ c 1 , and w ≤ c 1 , there is a non-negligible set Ω t,x,w ⊆ Ω such that ∆(ω) = 1, R(ω) = 1, and 1{S(ω) = j} = 1 when ω ∈ Ω t,x,w . If S 1 (θ 0 )(h) = 0 almost surely, then in particular, for almost every t ∈ [0, τ ], x ≤ c 1 , and w ≤ c 1 , S 1 (θ 0 )(h) = 0 when ω ∈ Ω t,x,w , which yields the following equation:

h Λj (t) + h β x + w h γj - K-1 k=1 w h γ k π k,γ0 (w) -e β 0 x t 0 h Λj (s) dΛ j,0 (s) + h β xΛ j,0 (t) = 0, (7)
with h γj = 0 when j = K. Then, by choosing t arbitrarily close to 0, and since Λ j,0 is continuous, Λ j,0 (0) = 0, and h Λj is continuous from the right at 0, we get that

h Λj (0) + h β x + w h γj - K-1 k=1 w h γ k π k,γ0 (w) = 0. ( 8
)
Taking the difference ( 7)-( 8) yields that

h Λj (t) -h Λj (0) = e β 0 x t 0 h Λj (s) dΛ j,0 (s) + h β xΛ j,0 (t) (9)
for almost every t ∈ [0, τ ] and x ≤ c 1 . Since Λ j,0 is increasing (by assumption (b)), for every t > 0, Λ j,0 (t) > Λ j,0 (0) = 0 and therefore (9) can be rewritten as

h Λj (t) -h Λj (0) Λ j,0 (t) = e β 0 x r(t) + h β x , (10)
where r(t) = t 0 h Λj (s) dΛ j,0 (s)/Λ j,0 (t). Consider first the case where β 0 = 0. Since the left-hand side of (10) does not depend on x, h β must equal 0. Next, consider the case where β 0 = 0. Let t 1 , t 2 > 0. Then e β 0 x [r(t 1 ) -r(t 2 )] does not depend on x. Since the covariance matrix of X is positive definite, we can find two distinct values x 1 and x 2 of X such that e β 0 x1 [r(t 1 ) -r(t 2 )] = e β 0 x2 [r(t 1 ) -r(t 2 )]. This implies that r(t 1 ) = r(t 2 ), from which we deduce that h Λj (t) has to be constant (say, equal to α) for almost every t ∈ (0, τ ]. From (10), we then deduce that h Λj (0) = α, which further implies that h β = 0, α = 0, and thus h Λj (t) = 0 for almost every t ∈ [0, τ ] (j ∈ K). This, together with (8) implies that h γj = 0, j ∈ K.

Let k = K. Then σ Λ K (h)(u) = P θ0 [1{S = K}φ(u, O, K, θ 0 )] h Λ K (u) = 0 for all u ∈ [0, τ ] since h β = 0 and h γ = 0. By assumptions (c) and (d), for every u ∈ [0, τ ] and k ∈ K,

P θ0 [1{S = k}φ(u, O, k, θ 0 )] = P θ0 1{S = k}Y (u)Q(O, k, θ 0 )e β 0 X ≥ P θ0 1{S = k}Y (τ )Re β 0 X > 0,
hence we conclude that h Λ K is identically equal to 0 on [0, τ ]. Next, considering σ Λ K-1 (h)(u) = 0 with h β = 0, h γ = 0 and h Λ K = 0, we conclude similarly that h Λ K-1 (u) = 0 for every u ∈ [0, τ ]. It follows that h Λj is identically equal to 0 on [0, τ ] for every j ∈ K. Therefore, σ is one-to-one.

We now turn to the proof of Theorem 3 itself. Similar to [START_REF] Fang | Maximum likelihood estimation in a semiparametric logistic/proportional-hazards mixture model[END_REF], we get that

√ n h β ( β n -β 0 ) + h γ ( γ n -γ 0 ) + K k=1 τ 0 h Λ k (s) d( Λ k,n -Λ k,0 )(s) = √ n S n (θ 0 )(σ -1 (h)) -P θ0 S 1 (θ 0 )(σ -1 (h)) + o p (1),
where S n is given by (3). Consider the subset {(h β , 0, 0; k ∈ K)|h β ∈ R p } ⊂ H and let h be an element of this subset. Setting h = h in the above equation yields

√ nh β ( β n -β 0 ) = √ n S n (θ 0 )(σ -1 ( h)) -P θ0 S 1 (θ 0 )(σ -1 ( h)) + o p (1). ( 11
)
By Lemma 2, the central limit theorem, and Slutsky's theorem,

√ nh β ( β n -β 0 )
is asymptotically normal with mean 0 and variance

P θ0 [S 1 (θ 0 )(σ -1 ( h)) 2 ]. If h ∈ H, direct calculation yields S 1 (θ 0 )(h) 2 = h β S β (θ 0 )S β (θ 0 ) h β + h γ S γ (θ 0 )S γ (θ 0 ) h γ + 2h β S β (θ 0 )S γ (θ 0 ) h γ +2h β S β (θ 0 ) K k=1 S Λ k (θ 0 )(h Λ k ) + 2h γ S γ (θ 0 ) K k=1 S Λ k (θ 0 )(h Λ k ) + K k=1 Q(O, k, θ 0 ) h Λ k (T )∆ -exp(β 0 X) T 0 h Λ k (s) dΛ k,0 (s) 2 +2 K k=1 j>k Q(O, k, θ 0 ) h Λ k (T )∆ -exp(β 0 X) T 0 h Λ k (s) dΛ k,0 (s) ×Q(O, j, θ 0 ) h Λj (T )∆ -exp(β 0 X) T 0 h Λj (s) dΛ j,0 (s) .
Taking expectation followed by some tedious algebraic manipulations and rearrangement of terms yield that

P θ0 S 1 (θ 0 )(h) 2 = h β σ β (h) + h γ σ γ (h) + K k=1 τ 0 σ Λ k (h)(u)h Λ k (u) dΛ k,0 (u). ( 12
) Therefore P θ0 S 1 (θ 0 )(σ -1 ( h)) 2 = σ -1 β ( h) σ β (σ -1 ( h)) + σ -1 γ ( h) σ γ (σ -1 ( h)) + K k=1 τ 0 σ -1 Λ k ( h)(u)σ Λ k (σ -1 ( h))(u) dΛ k,0 (u) = h β σ -1 β ( h),
where the last equality comes from the fact that

σ(σ -1 ( h)) = (σ β (σ -1 ( h)), σ γ (σ -1 ( h)), σ Λ k (σ -1 ( h)); k ∈ K) = h.
Now, recall that the linear map σ -1 β : R p → R p was defined in Section 4

as a restricted version of σ -1 β , by setting σ -1 β (h β ) = σ -1 β ( h) for any h of the form (h β , 0, 0; k ∈ K). Let {e 1 , . . . , e p } be the canonical basis of R p and Σ β = ( σ -1

β (e 1 ), . . . , σ -1 β (e p )). Then for any h β ∈ R p , we have σ

-1 β (h β ) = Σ β h β and thus P θ0 [S 1 (θ 0 )(σ -1 ( h)) 2 ] = h β Σ β h β . Hence, for every h β ∈ R p , √ nh β ( β n -β 0 ) converges in distribution to N (0, h β Σ β h β )
. By the Cramér-Wold device, √ n( β n -β 0 ) converges in distribution to N (0, Σ β ). We now turn to semiparametric efficiency of β n . Our proof relies on some geometric arguments. A detailed exposition of such arguments can be found in [START_REF] Bickel | Efficient and adaptive estimation for semiparametric models[END_REF] and [START_REF] Tsiatis | Semiparametric theory and missing data[END_REF] for example. Note that similar arguments have recently been used to derive efficiency in various other semiparametric hazard regression models (see [START_REF] Chang | An asymptotic theory for the nonparametric maximum likelihood estimator in the Cox gene model[END_REF] and Zeng et al. (2005) for example). For j = 1 . . . , p, denote h j = (e j , 0, 0; k ∈ K). Letting h = h j for each j = 1 . . . , p in turn in (11) yields

√ n( β n -β 0 ) = 1 √ n n i=1 l β (O i , θ 0 ) + o p (1),
where

l β (O, θ 0 ) = Σ β S β (θ 0 ) + ΞS γ (θ 0 ) + K k=1 S Λ k (θ 0 )(Ξ * ),
Ξ and Ξ * are (p × q) and (p × 1) matrices respectively defined by

Ξ =    σ -1 γ ( h 1 ) . . . σ -1 γ ( h p )    and Ξ * =    σ -1 Λ k ( h 1 ) . . . σ -1 Λ k ( h p )    ,
and S Λ k (θ 0 ) is applied componentwise to Ξ * . Thus β n is an asymptotically linear estimator for β 0 , and its influence function l β (O, θ 0 ) belongs to the tangent space spanned by the score functions. It follows that l β (O, θ 0 ) is the efficient influence function for β 0 , and that β n is semiparametrically efficient (see [START_REF] Bickel | Efficient and adaptive estimation for semiparametric models[END_REF] or [START_REF] Tsiatis | Semiparametric theory and missing data[END_REF]).

Remark. Note that l β (O, θ 0 ) is orthogonal to the space generated by the score functions for γ and the Λ k , k ∈ K. This can be ascertained as follows.

Let i ∈ {1, . . . , p}. The following hold:

e i Σ β = [σ -1 β ((e i , 0, 0; k ∈ K))] , e i Ξ = [σ -1 γ ((e i , 0, 0; k ∈ K))]
, and

e i S Λ k (θ 0 )(Ξ * ) = S Λ k (θ 0 )(σ -1 Λ k ((e i , 0, 0; k ∈ K))). Now, P θ0   e i l β (O, θ 0 ) K j=1 S Λj (θ 0 )(h Λj )   = P θ0   e i Σ β S β (θ 0 ) + ΞS γ (θ 0 ) + K k=1 S Λ k (θ 0 )(Ξ * )   K j=1 S Λj (θ 0 )(h Λj )     = P θ0 [{[σ -1 β ((e i , 0, 0; k ∈ K))] S β (θ 0 ) + [σ -1 γ ((e i , 0, 0; k ∈ K))] S γ (θ 0 ) + K k=1 S Λ k (θ 0 )(σ -1 Λ k ((e i , 0, 0; k ∈ K)))}( K j=1 S Λj (θ 0 )(h Λj ))] = P θ0 S 1 (θ 0 )(σ -1 ((e i , 0, 0; k ∈ K))) • S 1 (θ 0 )((0, 0, h Λj ; j ∈ K)) ,
where the last equality follows from equation (3). Now, similar calculations as the ones leading to (12) yield that

A.4 Proof of Theorem 4

The proof of asymptotic normality of √ n( γ n -γ 0 ) proceeds along the same line as for √ n( β n -β 0 ), and is therefore omitted. Next, for any t ∈ (0, τ ) and j ∈ K, the asymptotic normality of √ n( Λ j,n (t)-Λ j,0 (t)) can be proved by using a similar argument with h replaced by h (j,t) = (h β , h γ , h Λ k ; k ∈ K), where h β = 0, h γ = 0, h Λj (•) = 1{• ≤ t} (t ∈ (0, τ ) and j ∈ K), and h Λ k = 0 for every k ∈ K, k = j. Details are omitted.

A.5 Proof of Theorem 5

The proof of Theorem 5 parallels the proof of Theorem 3 in [START_REF] Parner | Asymptotic theory for the correlated gamma-frailty model[END_REF] and thus, will be kept brief. Let σ n = ( σ β,n , σ γ,n , σ Λ k ,n ; k ∈ K) be defined as σ with all of the θ 0 and P θ0 replaced by θ n and P n respectively. Similar to the proof of Theorem 3 in [START_REF] Parner | Asymptotic theory for the correlated gamma-frailty model[END_REF], it can be shown that σ n converges in probability to σ uniformly over H and that its inverse σ -1 n = ( σ -1 β,n , σ -1 γ,n , σ -1 Λ k ,n ; k ∈ K) is such that σ -1 n (h) converges to σ -1 (h) in probability.

For every h β , the asymptotic variance of √ nh β ( β n -β 0 ) is h β σ -1 β ((h β , 0, 0; k ∈ K)), which is consistently estimated by h β σ -1 β,n ((h β , 0, 0; k ∈ K)). Let h n = (h β,n , h γ,n , h Λ k ,n ; k ∈ K) = σ -1 n ((h β , 0, 0; k ∈ K)). Then σ n (h n ) = (h β , 0, 0; k ∈ K), or

   σ β,n (h n ) = h β σ γ,n (h n ) = 0 σ Λ k ,n (h n )(u) = 0, k ∈ K, u ∈ [0, τ ]. ( 13
)
In particular, letting u = T 1 , . . . , T n in (13) yields the following system of equations:

A n   h β,n h γ,n h Λ,n   =   h β 0 q 0 Kn   , (14)
where h Λ,n = (h Λ1,n (T 1 ), . . . , h Λ1,n (T n ), . . . , h Λ K ,n (T 1 ), . . . , h Λ K ,n (T n )) , and

A n is defined by (4). Some simple algebra on ( 14) yields that h β,n = Σ β,n h β where Σ β,n is defined in Section 4, and therefore h β Σ β,n h β is a consistent estimator of the asymptotic variance of √ nh β ( β n -β 0 ) for every h β . We conclude that Σ β,n converges in probability to Σ β . The consistency of Σ γ,n proceeds along the same lines and is therefore omitted.

We now turn to the estimation of the asymptotic variance of Λ j,n (t), for t ∈ (0, τ ) and j ∈ K. By the dominated convergence theorem and the consistency of σ -1 n , t 0 σ -1 Λj ,n (h (j,t) )(u) d Λ j,n (u) converges to v 2 j (t) = t 0 σ -1 Λj (h (j,t) )(u) dΛ j,0 (u), where we recall that h (j,t) is the element (h β , h γ , h Λ k ; k ∈ K) such that h β = 0, h γ = 0, h Λj (•) = 1{• ≤ t} for some t ∈ (0, τ ) and j ∈ K, and h Λ k = 0 for every k ∈ K, k = j. Letting h n = ( h β,n , h γ,n , h Λ k ,n ; k ∈ K) = σ -1 n (h (j,t) ), we get that σ n ( h n ) = h (j,t) or:

         σ β,n ( h n ) = 0 σ γ,n ( h n ) = 0 σ Λj ,n ( h n )(u) = 1{u ≤ t}, u ∈ [0, τ ] σ Λ k ,n ( h n )(u) = 0, k ∈ K, k = j, u ∈ [0, τ ]. (15)
In particular, letting u = T 1 , . . . , T n in (15) yields the system of equations

A n    h β,n h γ,n h Λ,n    =   0 p 0 q U n (j,t)   ,
with the notations h Λ,n = ( h Λ1,n (T 1 ), . . . , h Λ1,n (T n ), . . . , h Λ K ,n (T 1 ), . . . , h Λ K ,n (T n )) and U n (j,t) = (0 (j-1)n , 1{T 1 ≤ t}, . . . , 1{T n ≤ t}, 0 (K-j)n ) . Similar algebra as above yields

h Λ,n = Σ Λ,n U n (j,t) ,
where Σ Λ,n is defined in Section 4. Now, where Ξ n (j,t) = 0 (j-1)n , ∆Λ j,n (T 1 )1{T 1 ≤ t}, . . . , ∆Λ j,n (T n )1{T n ≤ t}, 0 (K-j)n . It follows that Ξ n (j,t) Σ Λ,n U n (j,t) is a consistent estimator of v 2 j (t), which concludes the proof.

h

  Λ k (u)σ Λ k (σ -1 ((e i , 0, 0; k ∈ K)))(u) dΛ k,0 (u) = 0.Thus, l β (O, θ 0 ) is orthogonal to the score K j=1 S Λj (θ 0 )(h Λj ). Similarly, l β (O, θ 0 ) is orthogonal to S γ (θ 0 ).

  Λj ,n (h (j,t) )(u) d Λ j,n (u) verifies Λj ,n (h (j,t) )(u) d Λ j,n (u) = Λj ,n (h (j,t) )(T i ) ∆Λ j,n (T i )1{T i ≤ t} = Ξ n (j,t) h Λ,n ,

		t 0 σ -1
	t	n
	σ -1	σ -1
	0	i=1

Acknowledgements

The authors gratefully acknowledge the helpful comments on an earlier version from the Associate Editor and a Referee.

Appendix A. Proofs of Theorems

A.1 Proof of Theorem 1

For every k ∈ K, define

denote the ordered failure times in this set. For any given sample size n, the NPML estimation method consists in maximizing L n in (2) over the approximating parameter space

• , L n achieves its maximum on this set.


Siwar Jendoubi
email: jendoubi.siwar@yahoo.fr

Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Ludovic Liétard
email: ludovic.lietard@univ-rennes1.fr

Boutheina Ben Yaghlane
email: boutheina.yaghlane@ihec.rnu.tn

Classification of message spreading in a heterogeneous social network

Keywords: Information propagation, heterogeneous social network, classification, evidence theory

Nowadays, social networks such as Twitter, Facebook and LinkedIn become increasingly popular. In fact, they introduced new habits, new ways of communication and they collect every day several information that have different sources. Most existing research works focus on the analysis of homogeneous social networks, i.e. we have a single type of node and link in the network. However, in the real world, social networks offer several types of nodes and links. Hence, with a view to preserve as much information as possible, it is important to consider social networks as heterogeneous and uncertain. The goal of our paper is to classify the social message based on its spreading in the network and the theory of belief functions. The proposed classifier interprets the spread of messages on the network, crossed paths and types of links. We tested our classifier on a real word network that we collected from Twitter, and our experiments show the performance of our belief classifier.

Introduction

Nowadays, social networks such as Twitter, Facebook and LinkedIn become increasingly popular. In fact, they introduced new habits and new ways of communication. Besides, one of the distinguishing features of on-line social networks is the information spreading through social links. This is due to the "wordof mouth" exchanges, i.e. user-to-user exchanges, which makes the information more accessible and it spreads and reaches a large scale in few minutes. The volume and the dynamic of the exchange has attracted the attention of research communities. This research is motivated by the fact that the study of the diffusion of information is useful for understanding the dynamic behind social networks and the evolution of human relationships. Thus, they have focused on the processing of such data to extract high quality information, this information may be an important event, it can also be useful for optimizing business performance, or even for preventing terrorist attacks, etc.

The processing of a social network, always, starts by studying its structural properties, in fact the simple visualization of the network cannot give us a clear analysis about it. In the literature, we found a lot of structural properties measures like the degree, the betweenness, the closeness, the eigenvector centrality, etc. Quantifying structural properties and interpreting them will be essential to characterize the behavior of social actors, their position in the network, their interactions and how do they diffuse the information. Hence, the analysis of the network structural properties is an essential step when we study and model information propagation.

In our work we are interested in the classification of the spreading of the information in a heterogeneous social network. We assume that each type of content has some specific behavior when it propagates in the network. Hence, we propose a new algorithm of information propagation in a heterogeneous social network that takes into account the behavior of the content to be propagated. Therefore, we introduce an evidential algorithm to classify the propagation of the information through the network.

In the next section, we outline the literature review of the information propagation in social networks, the social message classification and the theory of belief functions. In section three, we introduce our algorithm of information propagation in a heterogeneous social network. In section four, we present our classification algorithm. Finally, we present our experiments in the fifth section.

2 Literature review

Information propagation in social networks

Information dissemination is a wide research domain that attracted the attention of researchers from various field such as physics and biology. We find the family of epidemiological models that are used to understand how diseases spread through populations. The simplest version is SI (Suspected-Infected ), in this model, an individual is suspected if he has not the disease yet but he can catch it and become infected. This model was extended and many other version appeared to model specific diseases. Hence, we find SIS model (Suspected-Infected-Suspected ), SIR model (Suspected-Infected-Recovered ), SIRS model (Suspected-Infected-Recovered-Suspected ), etc. The reader can refer to [START_REF] Anderson | Infectious Diseases of Humans[END_REF][START_REF] Newman | Networks: An introduction[END_REF] for further details.

Computer scientists are generally interested in studying information propagation in on-line social networks. Mainly, their goal is to develop a model that simulates the diffusion process. Basic models are Linear Threshold Model (LTM) [START_REF] Granovetter | Threshold models of collective behavior[END_REF] and Independent Cascade Model (ICM) [START_REF] Goldenberg | Talk of the network: A complex systems look at the underlying process of word-of-mouth[END_REF]. They assume the existence of a structure of a directed graph where each node can be activated or not knowing that you can not inactivate already activated nodes. The ICM model requires a probability distribution which must be associated with each link and LTM requires a degree of influence that must be set on each link and a threshold of influence for each node [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. These two models were reused and improved in a lot of works like [START_REF] Galuba | Outtweeting the twitterers -predicting information cascades in microblogs[END_REF][START_REF] Saito | Learning diffusion probability based on node attributes in social networks[END_REF].

In this paper, we focus on information propagation in a heterogeneous social network, i.e. on which we find several types of links and/or nodes. In fact, in real word social networks we find many types of objects (users, groups, applications, etc) that are connected via many types of social links (friendship, membership, colleague, etc). Information dissemination in homogeneous social networks has been widely studied and the reader can refer to [START_REF] Guille | Information diffusion in online social networks: a survey[END_REF] for a recent survey. Now, research works start focusing on the processing of heterogeneous social networks. We find the work of [START_REF] Sermpezis | Information diffusion in heterogeneous networks: The configuration model approach[END_REF] that simulates the propagation of the information in heterogeneous social networks based on the configuration model approach. In [START_REF] Li | Influence propagation and maximization for heterogeneous social networks[END_REF], authors propose to consider the behavior of individuals to model the influence propagation, their model is based on a heterogeneous social network.

Social message classification

Social message classification approaches, presented in the literature, are generally based on the content of the information and text mining techniques. They search to classify the user generated content to positive or negative about a some specific product. This task is so called sentiment classification and it is used to mine opinions. It starts by an item and/or feature extraction step, then it compares the extracted items and/or features to an existing corpus, finally comes the sentiment classification that can be based on items, features or both of them [START_REF] Lo | A review of opinion mining and sentiment classification framework in social networks[END_REF]. We find the work of [START_REF] Mostafa | More than words: Social networks text mining for consumer brand sentiments[END_REF] in which the author used a random sample of 3516 tweets to classify the feelings of consumers with respect to well-known brands. He classified the opinions (tweets) into positive and negative to see what is the most dominant opinion. In [START_REF] He | Social media competitive analysis and text mining: A case study in the pizza industry[END_REF], a detailed case study that applies text mining to analyze unstructured textual content published on Twitter and Facebook and that talks about three chains of pizza. The reader can refer to the work of [START_REF] Nettleton | Survey data mining of social networks represented as graphs[END_REF] for a recent study of the state of the art of social networks data mining.

Theory of belief functions

Upper and Lower probabilities [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF] was the first ancestor of the theory of belief functions. Then comes the Mathematical theory of evidence [START_REF] Shafer | A mathematical theory of evidence[END_REF] which defines the basic framework of information management and processing in the evidence theory, often called Shafer model. The main purpose of the theory of belief functions is to achieve more reliable, precise and coherent information. Here we present a short introduction of this theory, for more details the reader can refer to [START_REF] Shafer | A mathematical theory of evidence[END_REF].

Let Ω = {ω 1 , ω 2 , . . . , ω n } be a set of all possible decisions that can be made in a particular problem, it is called frame of discernment. The basic belief assignment (BBA), m Ω , represents the agent belief on Ω, and it must respect A⊆Ω m Ω (A) = 1. In the case where we have m Ω (A) > 0, A is called focal set

Algorithm 1 Information propagation algorithm

Inputs:

-N: number of iteration -S: source of the message -Str: propagation strategy -Network: the heterogeneous social network Output:

-PrNet: propagation network Algorithm:

1 of m Ω . The basic belief assignment can be converted into other functions defined from 2 Ω to [0, 1]. This theory presents a rich framework for information fusion and combining pieces of information (evidence). We find the Dempster's rule [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF], the conjunctive and disjunctive combination rule [START_REF] Smets | Belief Functions: the Disjunctive Rule of Combination and the Generalized Bayesian Theorem[END_REF], etc.

Propagation algorithm in a heterogeneous social network

In this section we introduce an algorithm of information propagation in a heterogeneous social network. This new algorithm takes four different inputs which are the number of iterations (stopping condition), the source of the message, the propagation strategy and the heterogeneous social network. As output, we have the propagation network that preserves the traversed paths. Algorithm 1 shows outlines of our propagation process. It starts by the source node. First, we verify if the current node is ready (wants) to propagate the message. Then, for each type of link in the network we compute the number of neighbors that will receive the message. We assume that each type of message has some special characteristics of propagation in the network that is related to the types of links, so we define a propagation strategy for each type of message. Moreover, we consider the tendency of a particular node to propagate the message as a propagation parameter. Indeed, this parameter models the fact that a node can choose to distribute the message to a subset of its contacts (that he selects) or to retain it. The novelty of this algorithm is that we consider the type of the message while propagating it. Moreover our algorithm works with heterogeneous social networks where we have different types of links.

Classification of information propagation

The main purpose of this paper is to classify the spreading of the information through the network in order to characterize its content. In this section, we introduce our classification process that is composed of two steps; parameter learning step and the classification step. As mentioned in the algorithm 2, to learn the parameters of the model we need a set of propagation networks. First of all, we compute the number of nodes that have received the message via each type of link. We do this computation for each propagation level, i.e. we call propagation level the number of links between the source of the message and the target node. Second, we calculate the accrued effective by summing the effective of each level with the effective of the one before, this computation is done in order to preserve the propagation history at each propagation level. After that we transform the effective set of each level to a probability distribution defined on types of links, this transformation is done for two reasons; the first one, we need a probability distribution for the probabilistic classifier and the second one, it is an essential step to get the basic belief assignment distribution. Finally we transform each probability distribution to a BBA distribution using the consonant transformation [START_REF] Aregui | Consonant belief function induced by a confidence set of pignistic probabilities[END_REF][START_REF] Aregui | Constructing consonant belief functions from sample data using confidence sets of pignistic probabilities[END_REF].

Algorithm 2 Parameter learning algorithm

Input:

-PrNetSet: a set of propagation networks Output:

- Once model's parameters are learned, we can use it to classify new coming message (propagation network of the message) as shown in algorithm 3. Our classification algorithm starts by applying the same parameter learning process (algorithm 2) on the propagation network to be classified. Then for each level in the network we compute the distance between its probability distribution and the probability distribution of each propagation strategy, then we choose the class of the nearest propagation strategy (with the shortest distance) to be the class of the message in the current level. The same process is done with BBA distributions as mentioned in the algorithm.

Algorithm 3 Classification algorithm

Input:

-ProbaSets: a set of probabilities distributions for each strategy of propagation.

-BbaSets: a set of BBA distributions for each strategy of propagation.

-PrNet: The propagation network to be classified Output:

-In order to see the impact of the level of propagation on the classification results, in our output we have a class by level. Algorithm:

1. (ProbaPr, BbaPr) ← ParameterLearning (P rN et);

2. For i = 1 to NbrStrategies do (a) Foreach Level do i. ProbaDist(i, Level) ← Distance (ProbaPr, ProbaSets (i)); ii. BbaDist(i, Level) ← Distance (BbaPr, BbaSets (i)); 3. Foreach Level do (a)

Experiments and results

In this section, we present some experiments to show the power of the proposed evidential classification algorithm.

Data description

We used NodeXL V 1.0.1.245 [START_REF] Hansen | Analysing social media network with nodeXL insights from a connected world[END_REF] to collect social network data from Twitter. We collected the network shown in figure 1. It is a directed network in which nodes are Twitter users. Table 1 shows the characteristics of our network data.

As mentioned above, we need a heterogeneous social network to test proposed algorithms. Therefore, we used the structure of the network collected from Twitter and we generated, randomly, the types of links. We assume four types of link in the network which are "Professional", "Familial", "Friendly" and "Undefined". Then we obtained a heterogeneous social network that is used as input for our propagation algorithm.

Experiment configuration

In the following experiments, we defined three different propagation strategies for three types of messages which are: "Spam", "Professional" and "Familial". Each strategy is defined as the proportion of the nodes that will receive the message from each type of links. Hence, we have to define four proportions for each propagation strategy. To be as near as possible to the reality, we added a noise rate to the strategy. We note that the noise value can be added or removed from the proportions of kind of messages. We used the euclidean distance for the probabilistic classifier:

d E (P r 1 , P r 2 ) = card i=1 (P r 1 (i) -P r 2 (i)) 2 (1)
and the Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF] for the evidential one:

d J (m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D = (m 1 -m 2 ) (2)
such that D = is an 2 n × 2 n matrix and D (A, B) = |A∩B| A∪B . We fixed the number of levels in the network to three (three iterations in the propagation algorithm). Then we run the proposed propagation algorithm to create a training set for each propagation strategy, we fixed the size of the strategy training set to 100 propagation networks. Also, we created a testing set of size 100.

Results and discussion

In this section we present our results and a comparison between the probabilistic and the evidential classifier. To obtain accurate results we turned the experimental process ten times and we take the mean of the percentage of correctly classified (PCC) propagation networks. Figure 2 shows the impact of the propagation level on the PCC of the probabilistic results (figure 2a) and the evidential results (figure 2b). Figures 2a and2b illustrate that the PCC increases when the propagation level increases and we observe this fact starting from the noise level 20%. In figure 2a we observe that the curve of the second level coincides with the curve of third level and practically there is no improvement in the PCC. However, in figure 2b (evidential results), we note that the PCC increases with the propagation level, this fact is observed starting from the noise rate 20%. Hence, we have the PCC of the third level greater than the PCC of the first and the second levels, and the PCC of the second level is higher than the PCC of the first one. Therefore, more the message propagates in the network, more we can characterize it. In figure 3, we compare the probabilistic and the evidential results of the third propagation level. We note that without noise (0%) the probabilistic PCC is about 96% (with a 95% confidence interval of ±1.27) and the evidential PCC is equal to 93% (with a 95% confidence interval of ±1.60), but in real world social networks the absence of the noise is an ideal fact and cannot be realistic. When the noise rate increases, the curve shows that the percentage of correctly classified propagation networks (messages) decreases. However, we see that the evidential (Belief) PCC starts to be greater than the probabilistic (Proba) one. We observe this fact from the noise rate 20% where we have an evidential PCC equals to 70.7% (±4.33) and a probabilistic PCC equals to 65.8% (±4.18). Thus, we can conclude that the evidential classifier is more robust against the noise and gives better classification rates than the probabilistic classifier.

Conclusion

To conclude, we presented a state of the art of the information propagation, classification of social messages and the evidence theory. Then, we proposed an algorithm of information propagation in a heterogeneous social network. Thereafter we introduced a new evidential classification approach that classifies message propagation in a heterogeneous social network. Finally, we presented some experiments and we noticed the performance of the evidential classifier against the probabilistic one in noisy cases. Moreover, we observed that when the propagation level increases, the message class becomes more accurate and more realistic.

For future works, we will compare our propagation algorithm with previous algorithms. Also, we will search to improve it by the management of the uncertainty and the imprecision related to types of relationships between social actors. Our next goal is therefore to define a message propagation algorithm that takes into account the uncertainty of the types of relationships that is defined on the links, also we will search to consider the heterogeneity of nodes in the network. Second, we will run our classification algorithm with a more complex heterogeneous social network in order to prove its applicability.

  ProbaClasses(Level) ← StrategyMinDistance (ProbaDist (:, Level)); (b) BbaClasses(Level) ← StrategyMinDistance (BbaDist (:, Level));

Fig. 1 :

 1 Fig. 1: Network visualization

Fig. 2 :

 2 Fig. 2: The impact of the propagation level on the PCC

Fig. 3 :

 3 Fig. 3: Comparison between probabilistic results and evidential results (level three)

Table 1 :

 1 Data characteristics

	Vertices Edges	Geodesics	Betweenness Closeness Eigenvector
			distance
	97	350	6	184.99	0.004	0.01

the device MOBIDOC financed by the European Union under the PASRI program and administrated by the ANPR.