Jean-Marc Bardet
email: jean-marc.bardet@univ-paris1.fr

Kare Kamila
email: kamilakare@gmail.com

William Kengne
email: william.kengne@gmail.com

Consistent model selection criteria and goodness-of-fit test for affine causal processes

Keywords: MSC 2010 subject classifications: Primary 60K35, 60K35; secondary 60K35 model selection, affine causal processes, consistency, BIC, Portmanteau Test

This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR(∞) processes, as well as the GARCH or ARCH(∞), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. It appears from these conditions that the Bayesian Information Criterion (BIC) does not always guarantee the consistency. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on the portmanteau Test. Numerical simulations and an illustrative example on the FTSE index are performed to highlight the obtained asymptotic results, including a numerical evidence of the non consistency of the usual BIC penalty for order selection of an AR(p) models with ARCH(∞) errors.

Introduction

Model selection is an important tool for statisticians and all those who process data. This issue has received considerable attention in the recent literature. There are several model selection procedures, the main ones are : cross validation and penalized contrast based.

The cross validation ( [START_REF] Stone | Cross-validatory choice and assessment of statistical predictions[END_REF], [START_REF] Allen | The relationship between variable selection and data agumentation and a method for prediction[END_REF]) consists in splitting the data into learning sample, which will be used for computing estimators of the parameters and the test sample which allows to assess these estimators by evaluate their risks.

The procedures using penalized objective function search for a model, minimizing a trade-off between a sum of an empirical risk (for instance least squares, -2×log-likelihood), which indicates how well the model fits the data, and a measure of model's complexity so-called a penalty. The idea of penalizing dates back to the 1970s with the works of [START_REF] Mallows | Some comments on cp[END_REF] and [START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF]. By using the ordinary least squares in regression framework, Mallows obtained the C p criterion. Meanwhile, Akaike derived AIC for density estimation using log-likelihood contrast. A few years later, following Akaike, [START_REF] Schwarz | Estimating the dimension of a model[END_REF] proposed an alternative approach to density estimation and derived the Bayesian Information Criteria (BIC). The penalty term of these criteria is proportional to the dimension of the model. In the recent decades, different approaches of penalization have emerged such as the L 2 norm for the Ridge penalisation [START_REF] Hoerl | Ridge regression: Biased estimation for nonorthogonal problems[END_REF], the L 1 norm used by [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] that provides the LASSO procedure and the elastic-net that mixes the L 1 and L 2 norms [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF].

Model selection procedures can have two different objectives: consistency and efficiency. A procedure is said to be consistent if given a family of models, including the "true model", the probability of choosing the correct model approaches one as the sample size tends to infinity. On the other hand, a procedure is efficient when its risk is asymptotically equivalent to the risk of the oracle. In this work, we are interested to construct a consistent procedure for the general class of times series known as affine causal processes. This class of affine causal time series can be defined as follows. Let R ∞ be the space of sequences of real numbers with a finite number of non zero, if M , f : R ∞ → R are two measurable functions, then an affine causal class is Class AC(M, f ) : A process X = (X t ) t∈Z belongs to AC(M, f ) if it satisfies: X t = M (X t-i ) i∈N * ξ t + f (X t-i ) i∈N * for any t ∈ Z;

(1. [START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF] where (ξ) t∈Z is a sequence of zero-mean independent identically distributed random vectors (i.i.d.r.v) satisfying E(|ξ 0 | r ) < ∞ for some r ≥ 2 and E[ξ 2 0 ] = 1.

For instance,

• if M (X t-i ) i∈N * = σ and f (X t-i ) i∈N * = φ 1 X t-1 + • • • + φ p X t-p , then (X t ) t∈Z is an AR(p) process; • if M (X t-i ) i∈N * = a 0 + a 1 X 2 t-1 + • • • + a p X 2
t-p and f (X t-i ) i∈N * = 0, then (X t ) t∈Z is an ARCH(p) process. Note that, numerous classical time series models such as ARMA(p, q), GARCH(p, q), ARMA(p, q)-GARCH(p, q) (see [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF] and [START_REF] Ling | Asymptotic theory for a vector arma-garch model[END_REF]) or APARCH(δ, p, q) processes (see [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF]) belongs to AC(M, f ). The existence of stationary and ergodic solutions of this class has been studied in [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF] and [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF].

We consider a trajectory (X 1 , . . . , X n ) of a stationary affine causal process AC(M * , f * ), where M * and f * are unknown. We also consider a finite set M of parametric models m, which are affine causal time series. We assume that the "true" model m * corresponds to M * and f * . The aim is to obtain an estimator m of m * and testing the goodness-of-fit of the chosen model.

There already exist several important contributions devoted to the model selection for time series ; we refer to the book of [START_REF] Mcquarrie | Regression and Time Series Model Selection[END_REF] and the references therein for an overview on this topic. As we have pointed above, two properties are often used to evaluate a quality of a model selection procedure : consistency and efficiency. The first measure is often used when the true model is included in the collection of model's candidate ; otherwise, efficiency is the well-defined property. In many research in this framework, the main goal is to develop a procedure that fulfills one of these properties. So, in some classical linear time series models, the consistency of the BIC procedure has been established, see for instance [START_REF] Hannan | The estimation of the order of an arma process[END_REF] or [START_REF] Tsay | Order selection in nonstationary autoregressive models[END_REF] ; and the asymptotic efficiency of the AIC has been proved, see, among others, [START_REF] Shibata | Asymptotically efficient selection of the order of the model for estimating parameters of a linear process[END_REF], [START_REF] Hurvich | Regression and time series model selection in small samples[END_REF] for a corrected version of AIC for small samples, [START_REF] Ing | Order selection for same-realization predictions in autoregressive processes[END_REF], [START_REF] Ing | Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series[END_REF], [START_REF] Ing | Model selection for integrated autoregressive processes of infinite order[END_REF] for the case of infinite order autoregressive model. [START_REF] Shi | Regression model selection-a residual likelihood approach[END_REF] propose the (consistent) residual information criteria (RIC) for regression model (including regression models with ARMA errors) selection. In the framework of nonlinear threshold models, [START_REF] Kapetanios | Model selection in threshold models[END_REF] proved consistency results of a large class of information criteria, whereas [START_REF] Gao | Semiparametric non-linear time series model selection[END_REF] focussed on cross-validation type procedure for model selection in a class of semiparametric time series regression model. Let us recall that, the time series model selection literature is very extensive and still growing ; we refer to the monograph of [START_REF] Rao | On model selection[END_REF], which provided an excellent summary of existing model selection procedure, including the case of time series models as well as the recent review paper of [START_REF] Ding | Model selection techniques: An overview[END_REF].

The adaptive lasso, introduced by [START_REF] Zou | The adaptive lasso and its oracle properties[END_REF] for variable selection in linear regression models has been extended by [START_REF] Ren | Subset selection for vector autoregressive processes via adaptive lasso[END_REF] to vector autoregressive models, [START_REF] Kock | Consistent and conservative model selection with the adaptive lasso in stationary and nonstationary autoregressions[END_REF] carried out this procedure in stationary and nonstationary autoregressive models ; the oracle efficient is established. [START_REF] Lerasle | Optimal model selection for density estimation of stationary data under various mixing conditions[END_REF] considers model selection for density estimation under mixing conditions and derived oracle inequalities of the slope heuristic procedure ( [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] or [START_REF] Arlot | Data-driven calibration of penalties for least-squares regression[END_REF]) ; whereas [START_REF] Alquier | Model selection for weakly dependent time series forecasting[END_REF] develop oracle inequalities for model selection for weakly dependent time series forecasting. Recently, [START_REF] Shao | Oracally efficient estimation and consistent model selection for autoregressive moving average time series with trend[END_REF] have considered the model selection for ARMA time series with trend, and proved the consistency of BIC for the detrended residual sequence, while [START_REF] Arkoun | Sequential model selection method for nonparametric autoregression[END_REF] developed oracle inequalities of sequential model selection method for nonparametric autoregression. [START_REF] Hsu | On model selection from a finite family of possibly misspecified time series models[END_REF] pointed out that most existing model selection procedure cannot simultaneously enjoy consistency and (asymptotic) efficiency. They propose a misspecification-resistant information criterion that can achieve consistency and asymptotic efficiency for prediction using model selection.

In this paper, we focus on the class of models (1.1), and addressed the following questions :

imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

1. What regularity conditions are sufficient to build a consistent model selection procedure? Does the classic criterion such as BIC, still have consistent property for choosing a model among the collections M? 2. How can we test the goodness-of-fit of the chosen model? These questions have not yet been answered for the class of models and the framework considered here, in particular in case of infinite memory processes. This new contribution provides theoretical and numerical response of these issues.

(i) The estimator m of m * is chosen by minimizing a penalized criterion C(m) = -2 L n (m) + |m| κ n , where L n (m) is a Gaussian quasi-log-likelihood of the model m, |m| is the number of estimated parameters of the model m and κ n is a non-decreasing sequence of real numbers (see more details in Section 2). Note that, in the cases κ n = 2 or κ n = log n we respectively consider the usual AIC and BIC criteria. We provide sufficient conditions (essentially depending on the decreasing of the Lipschitz coefficients of the functions f and M ) for obtaining consistency of the model selection procedure. We also theoretically and numerically exhibit an example of order selection (weak AR(p) processes with ARCH(∞) errors) such that the consistency of the classical BIC penalty is not ensured.

(ii) We provide an asymptotic goodness-of-fit test for the selected model that is very simple to be used (with the usual Chi-square distribution limit), which successively completes the model selection procedure. Numerical applications show the accuracy of this test under the null hypothesis as well as an efficient test power under an alternative hypothesis. Note that, similar test has been proposed by [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF] under the Gaussian assumption on the observations, whereas [START_REF] Ling | Diagnostic checking of nonlinear multivariate time series with multivariate arch errors[END_REF] focused for multivariate time series with multivariate ARCH-type errors. Also, [START_REF] Duchesne | On diagnostic checking time series models with portmanteau test statistics based on generalized inverses and[END_REF] proposed a portmanteau test statistic based on generalized inverses and {2}-inverses for diagnostic checking in the class of model (1.1). Unlike these authors, we apply the test to a model obtained from a model selection procedure.

The paper is organized as follows. Some definitions, notations and assumptions are described in Section 2. The consistency of the criteria and the asymptotic normality of the post-model-selection estimator are studied in Section 3. In Section 4, the examples of AR(∞), ARCH(∞), AP ARCH(δ, p, q) and ARMA(p, q)-GARCH(p , q ) processes are detailed. The goodness-of-fit test is presented in Section 5. Finally, numerical results are presented in Section 6 and Section 7 contains the proofs.

Definitions and Assumptions

Let us introduce some definitions and assumptions in order to facilitate the presentation.

Notation and assumptions

In the sequel, we will consider a subset Θ of R d (d ∈ N). We will use the following norms:

• . denotes the usual Euclidean norm on R ν , with ν ≥ 1; • if X is R ν -random variable with r ≥ 1 order moment, we set X r = E( X r 1/r ; • for any set Θ ⊆ R d and for any g : Θ → R d , d ≥ 1, denote g Θ = sup θ∈Θ g(θ) .
In the introduction, to be more concise, we have presented the problem of time series model selection in a very general form. In reality, we will limit our field of study a little bit by considering a semiparametric framework. Hence, let (f θ ) θ∈Θ and (M θ ) θ∈Θ be two families of known functions such as for any θ ∈ Θ, both f θ , M θ with real values defined on R ∞ .

We begin by giving a condition on f θ and M θ which ensure the existence of a r-order moment, stationary and ergodic time series belonging to AC(M θ , f θ ). This condition, initially obtained in [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF], is written in terms of Lipschitz coefficients of both these functions. Hence, for Ψ θ = f θ or M θ , define:

Assumption A(Ψ θ , Θ): Assume that Ψ θ (0) Θ < ∞ and there exists a sequence of non-negative real numbers α k (Ψ θ , Θ) k≥1 such that ∞ k=1 α k (Ψ θ , Θ) < ∞ satisfying: Ψ θ (x) -Ψ θ (y) Θ ≤ ∞ k=1 α k (Ψ θ , Θ)|x k -y k | f or all x, y ∈ R ∞ .
Now for r ≥ 1, where ξ 0 r < ∞, define:

Θ(r) = θ ∈ R d , A(f θ , {θ}) and A(M θ , {θ}) hold with ∞ k=1 α k (f θ , {θ}) + ξ 0 r ∞ k=1 α k (M θ , {θ}) < 1 . (2.1)
Then, for any θ ∈ Θ(r), there exists a stationary and ergodic solution with r-order moment belonging to AC(M θ , f θ ). (see [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF] and [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]).

The framework

Let us start with an example to better understand the framework and the approach of model selection we will follow.

Example: Assume that the observed trajectory (X 1 , . . . , X n ) is generated from an AR(2) process and we would like to identify this family of process and its order. Then, we consider the collection M of ARMA(p, q) and GARCH(p , q ) processes for 0 ≤ p, q, p , q ≤ 9 and we would like to chose in this family a "best" model for fitting (X 1 , . . . , X n ). Note that there is 200 possible models and we expect to recognize the AR(2) as the selected model, at least when n is large enough.

We begin with the following property that allow to enlarge the family of models by extending the dimension d of the parameter θ:

Proposition 1. Let d 1 , d 2 ∈ N, Θ 1 ⊂ R d1 and Θ 2 ⊂ R d2 , and for i = 1, 2, define f (i) θi , M (i) θi : R ∞ → R and for θ i ∈ Θ i . Then there exist max(d 1 , d 2 ) ≤ d ≤ d 1 + d 2 , Θ ⊂ R d ,
and a family of functions f θ : R ∞ → R and M θ : R ∞ → [0, ∞) with θ ∈ Θ, such that for any θ 1 ∈ Θ 1 and θ 2 ∈ Θ 2 , there exists θ ∈ Θ satisfying AC M

(1)

θ1 , f (1) θ1
AC M

(2)

θ2 , f (2) θ2
⊂ AC M θ , f θ .

The proof of this proposition, as well as the other proofs, can be found in Section 7. This proposition says that it is always possible to embed two parametric causal affine models in a larger one. Hence, for instance, we can consider as well AR processes and ARCH processes in a unique representation, i.e.

               AR M (1) θ1 (X t-i ) i∈N * = σ f (1) θ1 (X t-i ) i∈N * = φ 1 X t-1 + • • • + φ p X t-p ARCH M (2) θ2 (X t-i ) i∈N * = a 0 + a 1 X 2 t-1 + • • • + a q X 2 t-q f (2) θ2 (X t-i ) i∈N * = 0 =⇒ M θ (X t-i ) i∈N * = θ 0 + θ 1 X 2 t-1 + • • • + θ q X 2 t-q f θ (X t-i ) i∈N * = θ q+1 X t-1 + • • • + θ q+p X t-p
.

From now and in all the sequel, we fix d ∈ N * , and the family of functions

f θ , M θ : R ∞ → R for θ ∈ Θ ⊂ Θ(r) ⊂ R d .
Let (X 1 , . . . , X n ) be an observed trajectory of an affine causal process X belonging to AC(M θ * , f θ * ), where θ * is an unknown vector of Θ, and therefore: In the sequel, we will consider several models, which all are particular cases of AC(M θ , f θ ) with θ ∈ Θ ⊂ R d . More precisely define:

X t = M θ * (X t-i ) i∈N * ξ t + f θ * (X t-i ) i∈N * for any t ∈ Z. ( 2
• a model m as a subset of {1, . . . , d} and denote |m| = #(m);

• Θ(m) = (θ i ) 1≤i≤d ∈ R d , θ i = 0 if i / ∈ m ∩ Θ;
• M as a family of models, i.e. M ⊂ P {1, . . . , d} .

Finally, for all m ∈ M, m ∈ AC(M θ , f θ ) when θ ∈ Θ(m) and denote m * the "true" model. We could as well consider hierarchical or exhaustive families of models.

Example: From the previous example, we can consider:

• a family M 1 of models m 1 such as M 1 = {1}, {1, 2}, . . . , {1, . . . , q + 1} : this family is the hierarchical one of ARCH processes with orders varying from 0 to q.

• a family M 2 of models m 2 such as M 2 = P {1, . . . , p + q + 1} : this family is the exhaustive one and contains as well the AR(2) process X t = φ 2 X t-2 + θ 0 ξ t as the process X

t = φ 1 X t-1 + φ 3 X t-3 + ξ t θ 0 + a 2 X 2 t-2 .
To establish the consistency of the selected model, we will need to assume that the "true" model m * with the parameter θ * , is included in the model family M.

The special case of NLARCH(∞) processes

As in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], in the special case of NLARCH(∞) processes, including for instance GARCH(p, q) or ARCH(∞) processes, a particular treatment can be realized for obtaining sharper results than using the previous framework. In such case, define the class:

Class AC( H θ ): A process X = (X t ) t∈Z belongs to AC( H θ ) if it satisfies:

X t = ξ t H θ (X 2 t-i ) i∈N * for any t ∈ Z. (2.3) Therefore, if M 2 θ (X t-i ) i∈N * = H θ (X t-i ) i∈N * = H θ (X 2 t-i ) i∈N * then, AC( H θ ) = AC(M θ , 0)
. In case of the class AC( H θ ), we will use the assumption A( H θ , Θ). By this way, we will obtain a new set of stationary solutions. For r ≥ 2 define:

Θ(r) = θ ∈ R d , A( H θ , {θ}) holds with ξ 0 r 2 ∞ k=1 α k ( H θ , {θ}) < 1 .
(2.4)

Then, for θ ∈ Θ(r), a process (X t ) t∈Z belonging to the class AC( H θ ) is stationary ergodic and satisfies X 0 r < ∞.

The Gaussian quasi-maximum likelihood estimation and the model selection criterion

In the sequel, for a model m ∈ M, a family of models of AC(M θ , f θ ) with θ ∈ Θ ⊂ R d , where θ → M θ and θ → f θ are two fixed functions, we are going to consider Gaussian quasi-maximum likelihood estimators (QMLE) of θ for each specific model m.

This approach as semi-parametric estimation has been successively introduced for GARCH(p, q) processes in [START_REF] Jeantheau | Strong consistency of estimators for multivariate arch models[END_REF] where its consistency is also proved, and the asymptotic normality of this estimator has been established in [START_REF] Berkes | GARCH processes: structure and estimation[END_REF] and [START_REF] Francq | Maximum likelihood estimation of pure garch and arma-garch processes[END_REF]. In [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], those results have been extended to affine causal processes, and an extension to Laplacian QMLE has been also proposed in [START_REF] Bardet | Asymptotic behavior of the laplacian quasi-maximum likelihood estimator of affine causal processes[END_REF]. The Gaussian QMLE is derived from the conditional (with respect to the filtration σ (X t ) t≤0 ) loglikelihood of (X 1 , . . . , X n ) when (ξ t ) is supposed to be a Gaussian standard white noise. Due to the linearity of a causal affine process, we deduce that this conditional log-likelihood (up to an additional constant) L n is defined for all θ ∈ Θ by:

L n (θ) := - 1 2 n t=1 q t (θ) , with q t (θ) := (X t -f t θ ) 2 H t θ + log(H t θ ) (2.5)
where

f t θ := f θ (X t-1 , X t-2 , • • • ), M t θ := M θ (X t-1 , X t-2 , • • • ) and H t θ = M t θ 2 .
Since L n (θ) depends on (X t ) t≤0 that are unknown, the idea of the quasi log-likelihood is to replace q t (θ) by an approximation q t (θ) and to compute θ as in equation (2.7) even if the white noise is not Gaussian. Hence, the conditional quasi log-likelihood (up to an additional constant) is given for all θ ∈ Θ by

L n (θ) := - 1 2 n t=1 q t (θ) , with q t (θ) := (X t -f t θ ) 2 H t θ + log( H t θ )
where

     f t θ := f θ (X t-1 , X t-2 , • • • , X 1 , u) M t θ := M θ (X t-1 , X t-2 , • • • , X 1 , u) H t θ := ( M t θ ) 2 (2.6)
for any deterministic sequence u = (u n ) with finitely many non-zero values (u = 0 is very often chosen without loss of generality).

However, the definitions of the conditional log-likelihood and quasi log-likelihood require that their denominators do not vanish. Hence, we will suppose in the sequel that the lower bound of

H θ (•) = M θ (•) 2 (which is reached since Θ is compact) is strictly positive: Assumption D(Θ): ∃h > 0 such that inf θ∈Θ (H θ (x)) ≥ h for all x ∈ R ∞ .
Finally, under this assumption, for each specific model m ∈ M, we define the Gaussian QMLE θ(m)

as θ(m) = argmax θ∈Θ(m) L n (θ). (2.7)
To select the "best" model m ∈ M, we chose a penalized contrast C(m) ensuring a trade-off between -2 times the maximized quasi log-likelihood, which decreases with the size of the model, and a penalty increasing with the size of the model. Therefore, the choice of the "best" model m among the estimated can be performed by minimizing the following criteria

m = argmin m∈M C(m) with C(m) = -2 L n θ(m) + |m| κ n , (2.8)
where

• (κ n ) n an increasing sequence depending on the number of observations n.

• |m| denotes the dimension of the model m, i.e. the cardinal of m, subset of {1, . . . , d}, which is also the number of estimated components of θ (the others are fixed to zero).

The consistency of the criterion C, i.e.

P( m = m * ) -→ n→∞ 1;
(2.9) will be established after showing that both of following probabilities are zero:

• the asymptotic probability of selecting a larger model containing the true model (overfitting case); • the asymptotic probability of selecting a false model that is a model not containing m * .

Asymptotic results

Assumptions required for the asymptotic study

The following classical assumption ensures the identifiability of the model considered.

Assumption Id(Θ):

For all θ, θ ∈ Θ, (f 0 θ = f 0 θ and M 0 θ = M 0 θ ) a.s. =⇒ θ = θ .
Another required assumption concerns the differentiability of Ψ θ = f θ or M θ on Θ. This type of assumption has already been considered in order to apply the QMLE procedure (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach[END_REF], [START_REF] White | Maximum likelihood estimation of misspecified models[END_REF]). First, the following Assumption Var(Θ) provides the invertibility of the "Fisher's information matrix" of X and is important to prove the asymptotic normality of the QMLE.

Assumption Var:

d i=1 α i ∂f 0 θ ∂θ (i) = 0 =⇒ ∀i = 1, . . . , d, α i = 0 a.s or d i=1 α i ∂H 0 θ ∂θ (i) = 0 =⇒ ∀i = 1, . . . , d, α i = 0 a.s .
Moreover, one of the following technical assumption is required to establish the consistency of the model selection procedure.

Assumption K(Θ): Assumptions A(f θ , Θ), A(M θ , Θ), A(∂ θ f θ , Θ), A(∂ θ M θ , Θ
) and B(Θ) hold and there exists r ≥ 2 such that θ * ∈ Θ(r). Moreover, with s = min(1, r/3), assume that the sequence

(κ n ) n∈N satisfies k≥1 ( 1 κ k ) s j≥k α j (f θ , Θ) + α j (M θ , Θ) + α j (∂ θ f θ , Θ) + α j (∂ θ M θ , Θ) s < ∞. Assumption K(Θ): Assumptions A( H θ , Θ), A(∂ θ H θ , Θ
) and B(Θ) hold and there exists r ≥ 2 such that θ * ∈ Θ(r). Moreover, with s = min(1, r/4), assume that the sequence

(κ n ) n∈N satisfies k≥1 ( 1 κ k ) s j≥k α j ( H θ , Θ) + α j (∂ θ H θ , Θ) s < ∞.
Remark 1. These conditions on (κ n ) n∈N have been deduced from conditions for strong law of large numbers obtained in [START_REF] Kounias | An inequality and almost sure convergence[END_REF] and are not too restrictive: for instance, if the Lipschitzian coefficients of f θ , M θ (the case using H θ can be treated similarly) and their derivatives are bounded by a geometric or Riemanian decrease:

1. the geometric case:

α j (f θ , Θ) + α j (M θ , Θ) + α j (∂ θ f θ , Θ) + α j (∂ θ M θ , Θ) = O(a j ) with 0 ≤ a < 1,
then any (κ n ) such as 1/κ n = o(1) can be chosen; for instance κ n = log n or log(log n); this is the case for instance of ARMA, GARCH, APARCH or ARMA-GARCH processes. 2. the Riemanian case:

α j (f θ , Θ) + α j (M θ , Θ) + α j (∂ θ f θ , Θ) + α j (∂ θ M θ , Θ) = O(j -γ ) with γ > 1: • if r ≥ 3 then -if γ > 2 then any sequence such as 1/κ n = o(1) can be chosen; -if 1 < γ < 2, any (κ n ) such as κ n = O(n δ ) with δ > 2 -γ can be chosen. • if 1 ≤ r < 3 -if γ > (r + 3)/r then any sequence such as 1/κ n = o(1) can be chosen; -if 1 < γ < (r + 3)/r then any (κ n ) such as κ n = n δ with δ > (r + 3)/r -γ can be chosen.
In the last case of these two conditions on r, we can see the usual BIC choice, κ n = log n does not fulfill the assumption in general.

New versions of limit theorems in [7]

These assumptions K(Θ) and K(Θ) used in Lemmas 1 and 2 (see Section 7) and the detailed Riemanian convergence rates of the previous remark, provide an improvement of the two main limit theorems established in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. More precisely, we obtain:

New version of Theorem 1 in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] Let (X 1 , . . . , X n ) be an observed trajectory of an affine causal process X belonging to AC(M θ * , f θ * ) (or AC( H θ )) where θ * is an unknown vector of Θ, a compact set included in

Θ(r) ⊂ R d (or Θ(r) ⊂ R d ) with r ≥ 2. Then, if assumptions A(f θ , Θ), A(M θ , Θ) (or A( H θ , Θ)), D(Θ), Id(Θ) hold with α j (f θ , Θ) + α j (M θ , Θ) = O j - for some > max(1 , 3/r) or α j ( H θ , Θ) = O j - for some > max(1 , 4/r) , (3.1
)

then the QMLE θ(m * ) satisfies θ(m * ) a.s. -→ n→+∞ θ * .
Proof. We use the same proof as in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] except for establishing

1 n L n (θ) -L n (θ) Θ a.s.
-→ n→+∞ 0. Indeed, we can apply Lemma 1 with κ n = n. Hence, this is checked under assumption K(Θ) under Riemanian condition of Remark 1 if r ≥ 3 when γ = > 1 and if 2 ≤ r ≤ 3, when γ = > 3/r, implying the first new conditions of the Theorem.

Under assumption K(Θ), an adaptation of Remark 1 implies that for r ≥ 4 we should have

γ = > 1 and if 2 ≤ r ≤ 4, when γ = > 4/r.
Therefore, in all the previous cases and when r = 4, we obtain a limiting decrease rate O j -γ with γ > 1 instead of γ > 3/2 in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. This can also be used to improve Theorem 2 in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]:

New version of Theorem 2 in [7]
If r ≥ 4 and under the assumptions of the previous new version of Theorem 1 in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], and Var(Θ), and if

assumptions A(∂ θ f θ , Θ), A(∂ θ M θ , Θ), A(∂ 2 θ 2 f θ , Θ) and A(∂ 2 θ 2 M θ , Θ) (or A(∂ θ H θ , Θ) and A(∂ 2 θ 2 H θ , Θ) ) hold with α j (∂ θ f θ , Θ) + α j (∂ θ M θ , Θ) = O j - or α j (∂ θ H θ , Θ) = O j - for some > 1, (3.2)
then the QMLE θ n (m * ) satisfies √ n θ(m * ) i -(θ * ) i i∈m * L -→ n→+∞ N |m * | 0 , F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 , (3.3)
with F (θ * , m * ) i,j = E ∂ 2 q 0 (θ * ) ∂θ i ∂θ j and (G(θ * , m * )) i,j = E ∂q 0 (θ * ) ∂θ i ∂q 0 (θ * ) ∂θ j for i, j ∈ m * .

Asymptotic model selection

Using the above assumptions, we can establish the limit theorem below, which provides sufficient conditions for the consistency of the model selection procedure.

Theorem 3.1. Let (X 1 , . . . , X n ) be an observed trajectory of an affine causal process X belonging to

AC(M θ * , f θ * ) (or AC( H θ )) where θ * is an unknown vector of Θ a compact set included in Θ(r) ⊂ R d (or Θ(r) ⊂ R d ) with r ≥ 4. If assumptions D(Θ), Id(Θ), K(Θ) (or K(Θ)), A(∂ 2 θ 2 f θ , Θ) and A(∂ 2 θ 2 M θ , Θ) (or A(∂ 2 θ 2 H θ , Θ)) also hold, then P( m = m * ) -→ n→∞ 1 and θ( m) P -→ n→∞ θ * . (3.4)
The following theorem shows the asymptotic normality of the QMLE of the chosen model.

√ n θ( m) i -(θ * ) i i∈m * L -→ n→+∞ N |m * | 0 , F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 , (3.5)
where F and G are defined in (3.3).

Remark 2.

In Remark 1, we detailed some situations where the assumption K(Θ) (or K(Θ)) holds, which leads to the results of Theorem 3.1 and 3.2. In particular, the log n penalty usually linked to BIC is consistent in the case of a geometric decrease of the Lipschitz coefficients of the functions f θ and M θ (and their first order derivative). In the case of a Riemanian rate, the consistency of BIC is not ensured; see also the next section.

Examples

In this section, some examples of time series satisfying the conditions of previous results are considered. These examples include AR(∞), ARCH(∞), AP ARCH(δ, p, q) and ARMA(p, q)-GARCH(p , q ).

AR(∞) models

For (ψ k (θ)) k∈N a sequence of real numbers depending on θ ∈ R d , let us consider an AR(∞) process defined by:

X t = k≥1 ψ k (θ * )X t-k + σ ξ t for any t ∈ Z, (4.1)
where (ξ t ) t admits 4-order moments, and

θ * ∈ Θ ⊂ Θ(4), the set of θ ∈ R d such that k≥1 ψ k (θ) Θ < 1 and σ > 0. This process corresponds to (2.2) with f θ (x i ) i≥1 = k≥1 ψ k (θ)x k and M θ ≡ σ for any θ ∈ Θ. The Lipschitz coefficients of f θ are α k (f θ ) = ψ k (θ) Θ . Moreover, Assumption D(Θ) holds with h = σ 2 > 0.
Let us consider M a finite family of models. Of course, the main example of such family of models is given by the one of ARMA(p, q) processes with 0 ≤ p ≤ p max and 0 ≤ q ≤ q max , providing (p max + 1)(q max + 1) models and θ ∈ R pmax+qmax+1 .

Besides, assume that Id(Θ), Var(Θ) hold and that the sequence

(ψ k ) is twice differentiable (with respect to θ) on Θ, with k ∂ 2 θ ψ k (θ) Θ < ∞ and ψ k (θ) Θ + ∂ θ ψ k (θ) Θ = O(k -γ ) with γ > 1. From Remark 1, • if γ > 2, the condition κ n -→ n→∞ ∞ (for instance, the BIC penalization with κ n = log(n), or κ n = √ n) ensures the consistency of m and the Theorem (3.2) holds if in addition θ * ∈ o Θ; • if 1 < γ < 2, κ n = O(n δ ) with δ > 2 -γ
has to be chosen (and we cannot insure the consistency of m in case of classical BIC penalization).

Finally, in the particular case of the family of ARMA processes, the stationarity condition implies that any κ n -→ n→∞ ∞ can be chosen (BIC penalization with κ n = log(n), or κ n = √ n), since the decreases of ψ k and its derivative are exponential.

ARCH(∞) models

For (ψ k (θ)) k∈N a sequence of nonnegative real numbers depending on θ ∈ R d , with ψ 0 > 0, let us consider an ARCH(∞) process defined by :

X t = ψ 0 (θ * ) + ∞ k=1 ψ k (θ * )X 2 t-k 1/2 ξ t for any t ∈ Z, (4.2)
where E ξ 4 0 < ∞, and

θ * ∈ Θ ⊂ Θ(4), the set of θ ∈ R d such that k≥1 ψ k (θ) Θ < 1. This process corresponds to (2.2) with f θ (x i ) i≥1 ≡ 0 and H θ (x i ) i≥1 = ψ 0 (θ) + ∞ k=1 ψ k (θ)x 2 k , i.e. H θ (y i ) i≥1 = ψ 0 (θ) + ∞ k=1 ψ k (θ)y k , for any θ ∈ Θ. The Lipschitz coefficients of H θ are α k ( H θ ) = ψ k (θ) Θ . Moreover, Assumption D(Θ) holds if h = inf θ∈Θ ψ 0 (θ) > 0.
Let us consider M a finite family of models. The main example of such family of models is given by the GARCH(p, q) processes with 0 ≤ p ≤ p max and 0 ≤ q ≤ q max , providing (p max + 1)(q max + 1) models and θ ∈ R pmax+qmax+1 . Moreover, assume that Id(Θ), Var(Θ) hold and that the sequence

(ψ k ) is twice differentiable (with respect to θ) on Θ, with k ∂ 2 θ ψ k (θ) Θ < ∞ and ψ k (θ) Θ + ∂ θ ψ k (θ) Θ = O(k -γ ) with γ > 1. From Remark
Θ; • if 1 < γ < 2, κ n = O(n δ
) with δ > 2 -γ has to be chosen (and we cannot insure the consistency of m in the case of the classical BIC penalization).

Finally, in the particular case of the family of GARCH processes, the stationarity condition implies that any κ n -→ n→∞ ∞ can be chosen (BIC penalization with κ n = log(n), or κ n = √ n), since the decreases of ψ k and its derivative are exponential.

4.3. APARCH(δ, p, q) models For δ ≥ 1 and from [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF], (X t ) t∈Z is an APARCH(δ, p, q) process with p, q ≥ 0 if:

X t = σ t ξ t (σ t ) δ = ω + p i=1 α i (|X t-i | -γ i X t-i ) δ + q j=1 β j (σ t-j ) δ for any t ∈ Z, (4.3)
where ω > 0, -1 < γ i < 1, α i ≥ 0, β j ≥ 0 for 1 ≤ i ≤ p and 1 ≤ j ≤ q, α p > 0, β q > 0 and q j=1 β j < 1. From [START_REF] Bardet | Asymptotic behavior of the laplacian quasi-maximum likelihood estimator of affine causal processes[END_REF], with θ = (ω, α 1 , . . . , α p , γ 1 , . . . , γ p , β 1 , . . . , β p ) , the conditional variance σ t can be rewritten as follows

σ δ t = b 0 (θ) + k≥1 b + k (θ)(max(X t-k , 0)) δ -b - k (θ)(min(X t-k , 0)) δ ; with f θ ≡ 0 and M t θ = σ t , we deduce that α k (M θ , Θ) = max( b + k (θ) 1/δ Θ , b - k (θ)
1/δ Θ ), and from the assumption q j=1 β j < 1, the Lipschitz coefficients α k (M θ , Θ) decrease exponentially fast. Then, the stationarity set for r ≥ 1 is

Θ(r) = θ ∈ R 2p+q+1 ξ 0 r ∞ j=1 max |b + j (θ)| 1/δ , |b - j (θ)| 1/δ < 1 .
Now, assume that (X t ) t∈Z is an APARCH(δ, p * , q * ) where 0 ≤ p * ≤ p max and 0 ≤ q * ≤ q max are unknown orders as well as the other parameters:

ω * > 0, -1 < γ * i < 1, α * i ≥ 0, β * j ≥ 0 for 1 ≤ i ≤ p max and 1 ≤ j ≤ q max , α p * > 0, β q * > 0.
Let M be the family of APARCH(δ, p, q) processes, with 0 ≤ p ≤ p max and 0 ≤ q ≤ q max . As a consequence, we consider here d = 2p max + q max + 1, and

θ * = t ω * , α * 1 , . . . , α * p * , 0, . . . , 0, γ * 1 , . . . , γ * p * , 0, . . . , 0, β * 1 , . . . , β * q * , 0, . . . , 0 ∈ R d .
With all the previous conditions, assumptions D(Θ), Id(Θ), Var(Θ) are satisfied. Moreover, since the Lipschitz coefficients decrease exponentially fast, K(Θ) is satisfied when κ n → ∞. Therefore, the consistency Theorem (3.1) and the Theorem (3.2) of the estimator of the chosen model are satisfied when r = 4 and κ n → ∞ (for instance with the typical BIC penalty κ n = log n).

4.4. ARMA(p, q)-GARCH(p , q ) models From [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF] and [START_REF] Ling | Asymptotic theory for a vector arma-garch model[END_REF], we define (X t ) t∈Z as an (invertible) ARMA(p, q)-GARCH(p , q ) process with p, q, p , q ≥ 0 if:

X t = p i=1 a i X t-i + ε t - q i=1 b i ε t-i ε t = σ t ξ t , with σ 2 t = c 0 + p i=1 c i ε 2 t-i + q i=1 d i σ 2 t-i for all t ∈ Z,
where Here we will consider the case of a stationary invertible ARMA(p, q)-GARCH(p , q ) process such as X 0 4 < ∞ and therefore we will consider:

• c 0 > 0, c p > 0, c i ≥ 0 for i = 1, • • • , p -1 and d q > 0, d i ≥ 0 for i = 1, • • • , q -1; • P (x) = 1 - p i=1 a i x i and Q(x) = 1 - q i=1 b i x i are
Θ p,q,p ,q = (a 1 , . . . , d q ) ∈ R p+q+p +1+q ,

q j=1 d j + ξ 0 4 p j=1 c j < 1 and 1 - p j=1 a j z j 1 - q j=1
b j z j = 0 for all |z| ≤ 1 .

Therefore, if (a 1 , . . . , d q ) ∈ Θ p,q,p ,q , (ε t ) t is a stationary GARCH(p , q ) process and (X t ) t is a stationary weak invertible ARMA(p, q) process. Moreover, following Lemma 2.1. of [START_REF] Bardet | Asymptotic behavior of the laplacian quasi-maximum likelihood estimator of affine causal processes[END_REF], we know that a stationary ARMA(p, q)-GARCH(p , q ) process is a stationary affine causal process with functions f θ and M θ satisfying the Assumption A(f θ , Θ) and A(M θ , Θ) with Lipschitzian coefficients decreasing exponentially fast, as well as their derivatives. Finally, if Θ is a bounded subset of Θ p,q,p ,q , then assumptions D(Θ), Id(Θ) and Var(Θ) are automatically satisfied.

Assume now that (X t ) t∈Z is an ARMA(p * , q * )-GARCH(p * , q * ) process where 0 ≤ p * ≤ p max , 0 ≤ q * ≤ q max , 0 ≤ p * ≤ p max and 0 ≤ q * ≤ q max are unknown orders with also unknown parameters:

c * 0 , . . . , c * p * , d * 1 , . . . , d * q * , a * 1 , . . . , a * p * , b * 1 , . . . , b q * .
Let M be the family of ARMA(p, q)-GARCH(p , q ) processes, with 0 ≤ p ≤ p max , 0 ≤ q ≤ q max , 0 ≤ p ≤ p max and 0 ≤ q ≤ q max . Hence, we consider here d = p max + q max + p max + q max + 1, and

θ * = c * 0 , . . . , c * p * , 0, . . . , 0, d * 1 , . . . , d * q * , 0, . . . , 0, a * 1 , . . . , a * p * , 0, . . . , 0, b * 1 , . . . , b q * , 0, . . . , 0 ∈ R d .
With Θ a bounded subset of Θ pmax,qmax,p max ,q max , all the previous assumptions D(Θ), Id(Θ), Var(Θ) are satisfied and K(Θ) is also satisfied as soon as κ n → ∞. As a consequence, in this framework the consistency Theorem (3.1) and the Theorem (3.2) of the estimator of the chosen model are satisfied when r = 4 and κ n → ∞ (for instance with the typical BIC penalty κ n = log n).

Portmanteau test

From the above section, we are now able to asymptotically pick up a best model in a family of models. We can also obtain asymptotic confident regions of the estimated parameter of the chosen model. However, it is also important to check whether the chosen model is appropriate. This section attempts to answer this question by constructing a portmanteau test as a diagnostic tool based on the squares of the residuals sequence of the chosen model. This test has been widely considered in the time series literature, with procedures based on the squared residual correlogram (see for instance [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF], [START_REF] Ling | Diagnostic checking of nonlinear multivariate time series with multivariate arch errors[END_REF] ) and the absolute residual (or usual residuals) correlogram (see for instance [START_REF] Li | On the asymptotic standard errors of residual autocorrelations in nonlinear time series modelling[END_REF], [START_REF] Duchesne | On diagnostic checking time series models with portmanteau test statistics based on generalized inverses and[END_REF], [START_REF] Li | Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity[END_REF]), among others. Since our goal is to provide an efficient test for the entire affine class that contains weak white noise processes, we consider in this setting the autocorrelation of the squared residuals and then we will follow the same scheme of procedure used in ( [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF], [START_REF] Ling | Diagnostic checking of nonlinear multivariate time series with multivariate arch errors[END_REF]) while relying on some of their results.

For m ∈ M, for K a positive integer, denote the vector of adjusted correlogram of squares residuals by: ρ(m) := ρ 1 (m), . . . , ρ K (m) ,

where for k = 1, . . . , K, ρ k (m) := γ k (m) γ 0 (m) with γ k (m) := 1 n n t=k+1 e 2 t (m) -1 e 2 t-k (m) -1 and e t (m) := M t θ(m) -1 X t -f t θ(m) .
Finally, the following theorem provides central limit theorems for ρ(m * ) and ρ( m) as well as for a portmanteau test statistic.

Theorem 5.1. Under the assumptions of Theorem 3.2, if

• E[ξ 3 0 ] = 0; • ∞ t=1 t -1/4 j≥t α j (f θ , Θ) + α j (M θ , Θ) 1/2 < ∞ or ∞ t=1 t -1/4 j≥t α j ( H θ , Θ) 1/2 < ∞; then, 1. With V (θ * , m * ) defined in (7.37), it holds that √ n ρ(m * ) L -→ n→+∞ N K 0 , V (θ * , m * ) .
(5.1)

2. With Q K (m * ) := n ρ(m * ) V ( θ(m * ), m * ) -1 ρ(m * ), we have Q K (m * ) L -→ n→+∞ χ 2 (K).
(5.2)

3. The previous points 1. and 2. also hold when m * is replaced by m.

Using the Theorem 5.1, we can asymptotically test:

   H 0 : ∃m * ∈ M, such as (X 1 , . . . , X n ) is a trajectory of X ∈ AC(M θ , f θ * ) with θ * ∈ Θ(m * ) H 1 : m * ∈ M, such as (X 1 , . . . , X n ) is a trajectory of X ∈ AC(M θ , f θ * ) with θ * ∈ Θ(m * )
.

Therefore, Q K ( m) can be used as a portmanteau test statistic to decide between H 0 and H 1 and diagnose the goodness-of-fit of the selected model.

Remark 3.

1. Like in [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF], it is important to point out that for ARCH(p) model, since

f t θ = 0, we have E ξ 2 0 -1 ∂ θ log M k θ *
= 0 for all k > p. Hence, for these models, the matrix V (θ * , m * )-I K

will have approximately zero entries from the (p + 1) th row onwards and then the standard errors of ρ(m * ) i are in this case equal to 1/ √ n for i = p + 1, . . . , K. The statistic 2 which will be asymptotically χ 2 distributed with K -p degrees of freedom. 2. In practice the constant µ 4 and the rows of the matrix V ( θ(m * ), m * ) involved in the previous theorem are estimated by the correspondent sample average; they are respectively µ 4 =

Q K (m * ) yields to Q(p, K) := n K i=p+1 [ ρ(m * ) i ]
1 n n t=1 ( e t ( m)) 4 and V ( θ(( m)), ( m)) k,. = 1 n n t=k+1 [( e t ( m)) 2 -1][∂ θ log M k θ )] (θ= θ( m)) .

Numerical Results

This section features some simulation experiments that are performed to assess the usefulness of the asymptotic results obtained in Section 3. The various configurations studied are presented below and we compare the performance of penalties log n and √ n. The process used to generate the trajectory is indicated each time.

Each model is generated independently 1000 times over a trajectory of length n. Different sample sizes are considered to identify possible discrepancies between asymptotically expected properties and those obtained at finite distance. We will consider n belongs to {100, 500, 1000, 2000}. Throughout this section, (ξ t ) represents a Gaussian white noise with variance unity.

Classical configurations

We first simulate some classical model illustrated as follows and the results are displayed in the Table 1.

1. Model 1, AR(2) process: X t = 0.4X t-1 + 0.4X t-2 + ξ t .
imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019 2. Model 2, ARMA(1, 1) process: X t = 0.3X t-1 + ξ t + 0.5ξ t-1 .

Model 3, ARCH(2) process: X

t = ξ t 0.2 + 0.4X 2 t-1 + 0.2X 2 t-2 .
We considered as competitive models all the models in the family M defined by: M = ARMA(p, q) or GARCH(p , q ) processes with 0 ≤ p, q, p ≤ 5, 1 ≤ q ≤ 5 .

As a consequence, there are 66 candidate models.

The Table 1 shows for each penalty (log n and √ n) the percentage of times the associated criterion selects respectively a wrong model, the true model and an overfitted model (here a model which contains the true model). From these results, it is clear that the consistency of our model selection procedure is numerically convincing, which is in accordance with Theorem 3.1, where both the criteria are consistent for Model 1, 2 and 3. Note also that the typical BIC log n penalty is the most interesting for retrieving the true model than the √ n-penalized likelihood for a small sample size. But the larger the sample size, the more accurate the √ n penalty case.

For each of the three models, we also applied the portmanteau test statistic Q K ( m), using the √ n penalty. Table 2 shows the empirical size and empirical power of this test. We call by empirical size, the percentage of falsely rejecting the null hypothesis H 0 . On the other hand, the empirical power represents the percentage of rejection of H 0 when we arbitrary chose a false model, which is a AR(3) process X t = 0.2X t-1 + 0.2X t-2 + 0.4X t-1 + ξ t for Model 1 and 2, and a ARCH(3) process

X t = ξ t 0.4 + 0.2X 2 t-1 + 0.2X 2 t-2 + 0.2X 2 t-3 for Model 3.
It is important to note that choosing the maximum number of lags K is sometimes tricky. To our knowledge, there is no real theoretical study to justify the choice of one value or another. However, some Monte Carlo simulations have suggested some ways to make a good choice . For instance [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF] suggested that the autocorrelations ρ k ( m) with 1 ≤ k ≤ K have a better asymptotic behaviour for small values of k. Therefore, the finite sample performance of the size and power of the test may also vary with the choice of K and could be better for small values of K. On the other hand, [START_REF] Tse | Testing for conditional heteroscedasticity: Some monte carlo results[END_REF] suggested that K = p + q + 1 may be an appropriate choice for the GARCH(p, q) family. Thus, in our tests, we consider K = 3 and K = 6 so that the rejection is based on the upper 5th percentile of the χ 2 (3) distribution on the one hand and χ 2 (6) on the other hand. Once again, the results of Table 2 numerically confirms the asymptotic results of Theorem 5.1. Remark that the test is more powerful by using values of K not too large as mentioned above especially for small samples.

Subset model selection

Now, we exhibit the performance of the criteria on a particular case of dimension selection. The process generated data is considered as follows:

Model 4 : X t = 0.4X t-3 + 0.4X t-4 + ξ t .

Here, we will consider the case of a nonhierarchical but exhaustive family M of AR(4) models , i.e.

M = P({1, 2, 3, 4}) =⇒ X t = θ 1 X t-1 + θ 2 X t-2 + θ 3 X t-3 + θ 4 X t-4 + ξ t and θ = (θ 1 , θ 2 , θ 3 , θ 4 ) ∈ Θ(m).
As a consequence, 16 = 2 4 candidate models are considered and Table 3 presents the results of the selection procedure. We deduce that the consistency of our model selection procedure is also numerically convincing in this case of exhaustive model selection, which is in accordance with Theorem 3.1

Slow decrease of the Lipschitz coefficients

In this subsection, we consider an AR(2) -ARCH(∞) with a slow decrease of its Lipschitz coefficients in order to numerically show that the penalty log n is not consistent in all cases. The considered data generating process is featured as follows:

Model 5 :

X t = -0.45 X t-1 + 0.4 X t-2 + ξ t with ξ t = ε t 0.5 + 0.1 i≥1 ξ 2 t-i /i 3 ,
where ε t is an i.i.d random sequence with mean 0 and variance 1. The sequence (α i ) i≥1 verifies α i = O(i -3 ) so that the sequence of Lipschitz coefficients of M ξ θ is given by α .5 ) and then the decrease rate of the sequence α i (M X θ ) is equal to O(i -1.5 ). From Remark 1, all penalties such as n δ with δ > 2 -1.5 = 0.5 will lead to a consistent model selection criterion and this is not the case for the typical BIC log n penalty. We have considered δ = 2/3 as in the Bridge Criteria (BC) recently proposed in [START_REF] Ding | Bridging aic and bic: a new criterion for autoregression[END_REF]. Here the family of model M is defined by M = AR(p)-ARCH(∞) processes with 1 ≤ p ≤ 8, where the ARCH(∞) is defined as in Model 5 .

i (M ξ θ ) = O(i -1
The results of simulations are featured in Table 4. Note that we also computed the log n criterion for n = 5000 and n = 10000 in additional numerical experiments and its frequencies of choice of the true order p = 2 were almost 72%. As a consequence, for this model selection framework of the infinite memory process with a slow decrease of Lipschitz coefficients, the usual BIC penalty log n seems numerically not sufficient to avoid overfitting in contrast with a n 2/3 penalty that leads to a consistent criterion.

Illustrative Example

We Therefore, the GARCH(p, q) family was considered for the modelling of the FTSE index with (p, q) ∈ 1; 10 × 0; 10 which lead us to 110 candidate models. The penalization log n and √ n have been applied to identify the best order and the goodness-of-fit of the selected model has been tested by the portmanteau test. Based on the results of the simulations, we set K = 3 for the portmanteau test statistic.

The GARCH(1, 1) is the "best" model according to both criteria (related to above penalizations) and the portmanteau statistic Q 3 ( m) 2.13 is associated with a p-value of 0.55. Hence, the selected model GARCH(1, 1) is adequate to model the FTSE 100 index using either criterion.

Proofs

We start with the proof of the Proposition 1.

Proof. For ease of writing, consider only the general case where f

(i) θi = g (i) αi and M (i) θi = N (i) βi where θ i = t (α i , β i ) for i = 1, 2. Now, assume that there exist α ∈ R δ , where 0 ≤ δ ≤ min(d 1 , d 2 ) and a function h α such as g (1) α1 = h α + (1) α 1 , f (2)
α2 = h α + (2) α 2 with α 1 = t (α, α 1 ) and α 2 = t (α, α 2 ) and (i) 0 = 0.
Similarly, assume that there exist β ∈ R δ , where 0 ≤ δ ≤ min(d 1 , d 2 ) and a function R β such as N

(1)

β1 = R β + m (1) β 1 , N (2)
β2 = R β + m (2) β 2 with β 1 = t (β, β 1 ) and β 2 = t (β, β 2 ) and m (i) 0 = 0. Consider now θ = t (α, α 1 , α 2 , β, β 1 , β 2 ) ∈ R d (and therefore max(d 1 , d 2 ) ≤ d ≤ d 1 + d 2 ), f θ = h α + (1) α 1 + (2) α 2 and M θ = R β + m (1) β 1 + m (2) β 2 . Then if X ∈ AC M θ , f θ , for any t ∈ Z, X t = R β ((X t-k ) k≥1 ) + m (1) β 1 ((X t-k ) k≥1 ) + m (2) β 2 ((X t-k ) k≥1 ) ξ t + h α ((X t-k ) k≥1 ) + (1) α 1 ((X t-k ) k≥1 ) + (2) α 2 ((X t-k ) k≥1 ) . Then, for α 2 = β 2 = 0, X ∈ AC M (1) θ1 , f (1) θ1
and for In the sequel, some lemmas are stated and theirs proofs are given.

α 1 = β 1 = 0, X ∈ AC M (2) θ2 , f (2)
Lemma 1. Let X ∈ AC(M θ , f θ ) (or AC( H θ )) and Θ ⊆ Θ(r) (or Θ ⊆ Θ(r)) with r ≥ 2.
Assume that the assumptions D(Θ) and K(Θ) (or K(Θ)) hold. Then:

1 κ n L n (θ) -L n (θ) Θ a.s.
-→ n→+∞ 0.

(7.1)

Proof. We have | L n (θ) -L n (θ)| ≤ n t=1 | q t (θ) -q t (θ)|. Then, 1 κ n L n (θ) -L n (θ) Θ ≤ 1 κ n n t=1 q t (θ) -q t (θ) Θ .
By Corollary 1 of [START_REF] Kounias | An inequality and almost sure convergence[END_REF], with r ≤ 3, (7.1) is established when:

k≥1 ( 1 κ k ) r/3 E q k (θ) -q k (θ) r/3 Θ < ∞. (7.2)
With r ≥ 3, and under the assumptions, we first recall some results already obtained in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]: for any t ∈ Z, Therefore, from (7.6), (7.7) and (7.4), there exists a constant C such that

• E |X t | r + f t θ r Θ + f t θ r Θ + M t θ r Θ + M t θ r Θ + H t θ r/2 Θ + H t θ r/2 Θ < ∞ (7.3) •          E f t θ -f t θ r Θ ≤ C j≥t α j (f θ , Θ) r E M t θ -M t θ r Θ ≤ C j≥t α j (M θ , Θ) r E H t θ -H t θ r/2 Θ ≤ C min j≥t α j (M θ , Θ) , j≥t α j (H θ , Θ) r/2 Θ ] 1/3 . ( 7
E ( q t (θ) -q t (θ) r/3 Θ ≤ C j≥t α j (f θ , Θ) + j≥t α j (M θ , Θ) r/3 . (7.8) Hence, k≥1 ( 1 κ
k ) r/3 E q k (θ) -q k (θ) r/3 Θ ≤ C k≥1 ( 1 κ k ) r/3 j≥k α j (f θ , Θ) + α j (M θ , Θ) r/3
, which is finite by assumption K(Θ), and this achieves the proof.

2/ If X ⊂ AC( H θ ) and using Corollary 1 of [START_REF] Kounias | An inequality and almost sure convergence[END_REF], with r ≤ 4, (7.1) is established when:

k≥1 ( 1 κ k ) r/4 E q k (θ) -q k (θ) r/4 Θ < ∞. (7.9)
By proceeding as in the previous case, we deduce

| q t (θ) -q t (θ)| ≤ h -2 |X t | 2 H t θ -H t θ Θ + h -1 H t θ -H t θ Θ .
In addition, we deduce that there exists a constant C such that

E ( q t (θ) -q t (θ) r/4 Θ ≤ C j≥t α j (H θ , Θ) r/4 . (7.10) Lemma 2. Let X ∈ AC(M θ , f θ ) (or AC( H θ )) and Θ ⊆ Θ(r) (or Θ ⊆ Θ(r)) with r ≥ 2.
Assume that the assumptions D(Θ) and K(Θ) (or K(Θ)) hold. Then:

1 κ n ∂ L n (θ) ∂θ - ∂L n (θ) ∂θ Θ a.s.
-→ n→+∞ 0.

Proof. We will go along similar lines as in the proof of Lemma 1. We have:

1 κ n ∂ L n (θ) ∂θ - ∂L n (θ) ∂θ Θ ≤ 1 κ n n t=1 ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i Θ .
Using again Corollary 1 of [START_REF] Kounias | An inequality and almost sure convergence[END_REF], it is sufficient to prove for r ≤ 3 that

k≥1 ( 1 κ k ) r/3 E ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i r/3 Θ < ∞. (7.12)
For any θ ∈ Θ, with

H θ = M 2 θ , the first partial derivatives of q t (θ) are -(H t θ ) -1 ∂H t θ ∂θ i . Then, using |a 1 b 1 c 1 -a 2 b 2 c 2 | ≤ |a 1 -a 2 | |b 2 | |c 2 |+|a 1 | |b 1 -b 2 | |c 2 |+|a 1 | |b 1 | |c 1 -c 2 | for any a 1 , a 2 , b 1 , b 2 , c 1 , c 2 in R, we obtain Θ ∂H t θ ∂θ i Θ .
Using again the results of [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], we know that:

• E ∂f t θ ∂θ i r Θ + ∂ f t θ ∂θ i r Θ + ∂M t θ ∂θ i r Θ + ∂ M t θ ∂θ i r Θ + ∂H t θ ∂θ i r/2 Θ + ∂(H t θ ) -1 ∂θ i r Θ < ∞ (7.13) •                                E (H t θ ) -1 -( H t θ ) -1 r Θ ≤ C j≥t α j (M θ , Θ) r E ∂f t θ ∂θ i - ∂ f t θ ∂θ i r Θ ≤ C j≥t α j (∂f θ , Θ) r E ∂H t θ ∂θ i - ∂ H t θ ∂θ i r/2 Θ ≤ C j≥t α j (M θ , Θ) + α j (∂M θ , Θ) r/2 E ∂(H t θ ) -1 ∂θ i - ∂( H t θ ) -1 ∂θ i r/2 Θ ≤ C j≥t α j (M θ , Θ) + α j (∂M θ , Θ) r/2 (7.14)
1. If X ⊂ AC(M θ , f θ ), we deduce from the Hölder's Inequality that,

E ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i r/3 Θ ≤ C E f t θ -f t θ r Θ 1/3 E ∂ f t θ ∂θ i r/2 Θ 2/3 + E X t -f t θ 2r/3 Θ 1/2 E ∂f t θ ∂θ i - ∂ f t θ ∂θ i r Θ 1/3 + E (H t θ ) -1 -( H t θ ) -1 r Θ 1/3 E X t -f t θ r Θ E ∂ f t θ ∂θ i r Θ 1/3 + E X t -f t θ r Θ 1/3 E ∂( H t θ ) -1 ∂θ i - ∂(H t θ ) -1 ∂θ i r/2 2/3 + E ∂(H t θ ) -1 ∂θ i r Θ 1/3 E |X t | r E f t θ -f t θ r Θ 1/3 + E ∂ H t θ ∂θ i - ∂H t θ ∂θ i r/3 Θ + E ∂H t θ ∂θ i r/2 Θ 2/3 E ( H t θ ) -1 -(H t θ ) -1 r Θ 1/3
. Using (7.13) and (7.14), we deduce Therefore,

E ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i r/3 Θ ≤ C j≥t α j (f θ , Θ) + α j (M θ , Θ) + α j (∂f θ , Θ) + α j (∂M θ , Θ)
k≥1 1 κ r/3 k E ∂ q k (θ) ∂θ i - ∂q k (θ) ∂θ i r/3 Θ ≤ C k≥1 1 κ r/3 k j≥t α j (f θ , Θ) + α j (M θ , Θ) + α j (∂f θ , Θ) + α j (∂M θ , Θ) r/3
.

We conclude the proof of (7.12) from assumption K(Θ).

2. If X ⊂ AC( H θ ), we deduce

∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i Θ ≤ X t 2 ∂( H t θ ) -1 ∂θ i - ∂(H t θ ) -1 ∂θ i Θ + h -1 ∂ H t θ ∂θ i - ∂H t θ ∂θ i Θ + ( H t θ ) -1 -(H t θ ) -1 Θ ∂H t θ ∂θ i Θ .
As a consequence,

E ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i r/4 Θ ≤ E X t r E ∂( H t θ ) -1 ∂θ i - ∂(H t θ ) -1 ∂θ i r/2 Θ 1/2 + h -r/4 E ∂ H t θ ∂θ i - ∂H t θ ∂θ i r/4 Θ + E ( H t θ ) -1 -(H t θ ) -1 r/2 Θ E ∂H t θ ∂θ i r/2 Θ 1/2 , implying E ∂ q t (θ) ∂θ i - ∂q t (θ) ∂θ i r/4 Θ ≤ C j≥t α j (H θ , Θ) + α j (∂H θ , Θ) r/4
, which achieves the proof, according to Corollary 1 of [START_REF] Kounias | An inequality and almost sure convergence[END_REF]. Proof. We have:

1 κ n L n ( θ(m)) -L n (θ * ) = 1 κ n L n ( θ(m)) -L n ( θ(m)) + L n ( θ(m)) -L n (θ * ) + L n (θ * ) -L n (θ * ) ≤ 2 κ n L n (θ) -L n (θ) Θ(r) + 1 κ n L n ( θ(m)) -L n (θ * ) . According to Lemma 1, 1 κn L n (θ) -L n (θ) Θ(r)
a.s.

-→ n→+∞ 0. The proof will be achieved if we prove

1 κ n L n ( θ(m)) -L n (θ * ) = o P (1). (7.16)
Applying a second order Taylor expansion of L n around θ(m) for n sufficiently large such that θ(m) ∈ Θ(m) which are between θ(m) and θ * , yields: Let us deal first with the first term on the right hand side of last equality:

1 κ n L n ( θ(m)) -L n (θ * ) = 1 κ n θ(m) -θ * ∂L n ( θ(m)) ∂θ + 1 2κ n θ(m) -θ * ∂ 2 L n (θ(m)) ∂θ 2 θ(m) -θ * . (7.
1 κ n θ(m) -θ * ∂L n ( θ(m)) ∂θ = 1 κ n √ n θ(m) -θ * 1 √ n ∂L n ( θ(m)) ∂θ .
Since 1 κn = o(1) and from [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] we have On the other hand, for the second term of the right hand side of equality (7.17), let us note that, we have from [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]:

√ n θ(m) -θ * = O P (1)
• √ n θ(m) -θ * L -→ n→+∞ A θ * ,m a Gaussian random variable from (3.3). • - 2 n ∂ 2 L n (θ(m)) ∂θ i ∂θ j i,j∈m a.s. -→ n→+∞ F (θ * , m) since θ(m) a.s.
-→ n→+∞ θ * and using the assumption Var(Θ)

insuring that the matrix F (θ * , m) exists and is definite positive (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]).

Hence,

θ(m) -θ * ∂ 2 L n (θ(m)) ∂θ i ∂θ j i,j∈m ( θ(m) -θ * ) = -1 2 √ n θ(m) -θ * F (θ * , m) + o P (1) √ n θ(m) -θ * P -→ n→∞ -1 2 A θ * ,m F (θ * , m) A θ * ,m .
We deduce that

θ(m) -θ * ∂ 2 L n (θ(m)) ∂θ i ∂θ j i,j∈m ( θ(m) -θ * ) = O P (1) =⇒ 1 κ n θ(m) -θ * ∂ 2 L n (θ(m)) ∂θ i ∂θ j i,j∈m
( θ(m) -θ * ) = o P (1). (7.19) Thus, (7.16) follows from (7.17), (7.18) and (7.19); which completes the proof of Lemma 3.

Misspecified model

When a model m is misspecified, we will show that P( m = m * ) -→ n→∞ 0 following the same scheme of proof than in [START_REF] Sin | Information criteria for selecting possibly misspecified parametric models[END_REF]. Before dealing with this proof, we state some useful results.

Proposition 2. Let X ∈ AC(M θ , f θ ) (or AC( H θ )) and Θ ⊆ Θ(r) (or Θ ⊆ Θ(r)) with r ≥ 2. Then, when the assumption D(Θ) holds,

1 n L n (θ) -L(θ) Θ a.s. -→ n→+∞ 0 with L(θ) := - 1 2 E[q 0 (θ)]. (7.20)
Proof. See the proof of Theorem 1 in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. Proof. Applying a second order Taylor expansion of L n around θ(m * ) for n sufficiently large such that θ(m) ∈ Θ(m) which are between θ * and θ(m * ), yields:

L n ( θ(m)) -L n (θ * ) = ( θ(m) -θ * ) ∂L n ( θ(m)) ∂θ + 1 2 ( θ(m) -θ * ) ∂ 2 L n (θ(m)) ∂θ∂θ ( θ(m) -θ * ) = √ n( θ(m) -θ * ) 1 √ n ∂L n ( θ(m)) ∂θ + 1 2 √ n( θ(m) -θ * ) 1 n ∂ 2 L n (θ(m)) ∂θ∂θ √ n( θ(m) -θ * ) = o p (
P( m = m) ≤ P C(m) ≤ -2 L n θ * + |m * | κ n ≤ P -2 L n ( θ) -L n (θ * ) ≤ κ n (|m * | -|m|) ≤ P 1 κ n L n (θ * ) -L n ( θ) ≤ (|m * | -|m|) 2 -→
L n ( θ(m * )) -L n ( θ(m)) = L n ( θ(m * )) -L n ( θ(m * )) -L n ( θ(m)) -L n ( θ(m)) + L n ( θ(m * )) -L n ( θ(m)) . (7.22)
It follows from Lemma 1 that the first and the second term of the right part of (7.22) are equal to o P (κ n ). Moreover, the third term can be written as follows:

L n ( θ(m * )) -L n ( θ(m)) = L n ( θ(m * )) -L n (θ * ) + L n (θ * ) -L n ( θ(m)) .
L n (θ * ) -L n (θ) = L n (θ * ) -n L(θ * ) -L n (θ) -n L(θ) + n L(θ * )) -L(θ)) = o P (n) + n L(θ * )) -L(θ) .
Let us denote by F t := σ X t-1 , X t-2 , • • • . Using conditional expectation, we obtain

L(θ * ) -L(θ) = - 1 2 E E q 0 (θ) -q 0 (θ * ) | F 0 . (7.24)
= 1. Then A(m) > 0.

Therefore A(m) > 0 for any θ ∈ Θ(m) and particularly for θ = θ(m) and (7.23) holds. Thus, (7.22) yields to

L n ( θ(m * )) -L n ( θ(m)) = o P (κ n ) + O P (1) + n A(m) + o P (n) = O P (1) + n A(m) + o P (n).
Finally, when m ∈ M such as m * ⊂ m, we have

C(m) -C(m * ) = 2 n A(m) + o P (n) + O P (1) + κ n (|m| -|m * |) P -→ n→∞ + ∞ since κ n = o(n), therefore P C(m) > C(m * ) -→ n→∞ 1.
Thus we have proved the first and most difficult part of Theorem (3.1). The next lines show the second part which is about the consistency of θ( m).

Given > 0, we have :

P θ( m) -θ * i∈m * > = P θ( m) -θ * i∈m * > | m = m * P m = m * +P θ( m) -θ * i∈m * > | m = m * P m = m * .
From the strong consistency of the QMLE (see New version of Theorem 1 of [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]), the first term of the right hand side of the above equation is asymptotically zero and also the second one under the assumptions of the first part of Theorem 3.1 which gives P m = m * -→ n→∞ 0.

7.3. Proof of Theorem 3.2

Proof. For x = (x i ) 1≤i≤d ∈ R d , denote F n (x) = P 1≤i≤d √ n θ( m) -θ * i ≤ x i .
First, we have:

F n (x) = P 1≤i≤d √ n θ( m) -θ * i ≤ x i m = m * P m = m * +P 1≤i≤d √ n θ( m) -θ * i ≤ x i m = m * P m = m * .
Under the assumptions of Theorem

P 1≤i≤d √ n θ( m) -θ * i ≤ x i m = m * = P i∈m * √ n θ(m * ) -θ * i ≤ x i i / ∈m * √ n θ(m * ) -θ * i ≤ x i . Since θ(m * ) ∈ Θ(m * ), θ(m * ) i i / ∈m * = θ * i i / ∈m * = 0, for (x i ) i /
∈m * a family of non negative real numbers we have:

P i∈m * √ n θ(m * ) -θ * i ≤ x i i / ∈m * √ n θ(m * ) -θ * i ≤ x i = P i∈m * √ n θ(m * ) -θ * i ≤ x i -→ n→∞ P F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 -1/2 Z ≤ (x i ) i∈m * ,
with Z a standard Gaussian random vector in R |m * | from the central limit theorem (3.3), and this achieves the proof of 3.5 of Theorem 3.2.

Proof of Theorem 5.1

Consider the following notation: for θ ∈ Θ and m ∈ M, denote the residuals and quasi-residuals by:

e t (θ) := M t θ -1 X t -f t θ and e t (θ) := M t θ -1 X t -f t θ e t (m) := M t θ(m) -1 X t -f t θ(m)
and e t (m) :

= M t θ(m) -1 X t -f t θ(m)
.

For k ∈ {0, 1, . . . , n -1}, θ ∈ Θ and m ∈ M, define also the adjusted lag-k covariograms and correlograms of the squared (standardized) residual by:

           γ k (θ) := 1 n n-k t=1 e 2 t (θ) -1 e 2 t+k (θ) -1 and γ k (θ) := 1 n n-k t=1 e 2 t (θ) -1 e 2 t+k (θ) -1 γ k (m) := 1 n n-k t=1 e 2 t (m) -1 e 2 t+k (m) -1 and γ k (m) := 1 n n-k t=1 e 2 t (m) -1 e 2 t+k (m) -1 and ρ k (θ) := γ k (θ) γ 0 (θ) , ρ k (θ) := γ k (θ) γ 0 (θ) , ρ k (m) := γ k (m) γ 0 (m) and ρ k (m) := γ k (m) γ 0 (m) .
Finally, for K a positive integer, denote the vector of adjusted correlogram: ρ(θ) := ρ 1 (θ), . . . , ρ K (θ) and ρ(m) := ρ 1 (m), . . . , ρ K (m) .

Proof.

(1) This proof is divided into two parts. In (i) we prove a result that ensures that the asymptotic distributions of the vectors ρ(θ) and ρ(θ) are the same. In (ii) we show that the large sample distribution of √ nρ(m * ) is normal with a covariance matrix V (θ * , m * ) . Those two conditions do lead well to the asymptotic normality (5.1).

(i) In this part, we first show that for any k ∈ N,

√ n γ k (θ) -γ k (θ) Θ a.s. -→ n→∞ 0. (7.25)
We have: Now, we show that I 1 Θ a.s.

√ n γ k (θ) -γ k (θ) = 1 √ n n t=k+1 e 2 t (θ) -1 e 2 t-k (θ) -1 - 1 √ n n t=k+1 e 2 t (θ) -1 e 2 t-k (θ) -1 = 1 √ n n t=k+1 e 2
-→ n→+∞ 0. We can rewrite I 1 as follows

I 1 = 1 √ n n t=k+1 e 2 t-k (θ) e 2 t (θ) -e 2 t (θ) + 1 √ n n t=k+1 e 2 t (θ) e 2 t-k (θ) -e 2 t-k (θ) = 1 √ n n t=k+1 e 2 t-k (θ) -e 2 t-k (θ) e 2 t (θ) -e 2 t (θ) + 1 √ n n t=k+1 e 2 t-k (θ) e 2 t (θ) -e 2 t (θ) + 1 √ n n t=k+1 e 2 t (θ) e 2 t-k (θ) -e 2 t-k (θ) := I 1 1 + I 2 1 + I 3 1 .
Let us show that I 1 1 Θ a.s.

-→ n→+∞ 0 in our two frameworks.

a/ If X ⊂ AC(M θ , f θ ), by Hölder's inequality, it follows from (7.8) that,

E e 2 t-k (θ) -e 2 t-k (θ) e 2 t (θ) -e 2 t (θ) 1/2 Θ ≤ E e 2 t (θ) -e 2 t (θ) Θ × E e 2 t-k (θ) -e 2 t-k (θ) Θ 1/2 × E M t θ -M t θ 2 Θ 1/2 ≤ C E j≥t α j (f θ , Θ)X t-j 2 1/2 + C E j≥t α j (M θ , Θ)X t-j 2 1/2 ≤ C j≥t α j (f θ , Θ) + α j (M θ , Θ), using E |X t | 4 + f t θ 2 Θ + f t θ 2 Θ < ∞ and Cauchy-Schwarz Inequality. Hence, E e 2 t-k (θ) -e 2 t-k (θ) e 2 t (θ) -e 2 t (θ) 1/2 Θ ≤ C j≥t-k α j (f θ , Θ) + α j (M θ , Θ).
Therefore, from [START_REF] Kounias | An inequality and almost sure convergence[END_REF], which is also the condition for insuring that I 2 Θ a.s.

I 1 1 Θ a.s. -→ n→+∞ 0 when ∞ t=1 t -1/4 j≥t α j (f θ , Θ) + α j (M θ , Θ) < ∞. ( 7
-→ n→+∞ 0 and I 3 Θ a.s.

-→ n→+∞ 0. This ends the proof of (7.25). Finally, since ρ k (θ) = γ k (θ)/ γ 0 (θ) and ρ k (θ) = γ k (θ)/γ 0 (θ), with γ 0 (θ) > 0, we deduce under condition (7.28) that √ n ρ k (θ) -ρ k (θ) Θ a.s.

-→ n→+∞ 0 for any k ≥ 1. (7.29) This also implies √ n ρ k (m * ) -ρ k (m * ) a.s.

-→ n→+∞ 0 for any k ≥ 1.

(7.30)

(ii) The proof of this result has already been done in [START_REF] Li | On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity[END_REF] but in a Gaussian framework. We recall here the main lines while avoiding the Gaussian assumption. The first step is to use a Taylor expansion of the function γ. Hence, we have for each k = 1, . . . , K,

√ n γ k (m * ) = √ n γ k ( θ(m * )) = √ n γ k (θ * ) + ∂ θ γ k (θ (k) ) √ n ( θ(m * )) i -θ * i i∈m * , (7.31)
where ∂ θ γ k = t ∂γ k /∂θ i i∈m * , and θ (k) is in the ball of centre θ * and radius ( θ(m * ) -θ * ) i∈m * . We also have

∂ θ γ k (θ) = - 2 n
sure convergence occurs for ∂ θ γ k (θ (k) ). Then, we finally obtain

∂ θ γ k (θ (k) ) 1≤k≤K a.s. -→ n→+∞ J K (m * ) = -2 E ξ 2 0 -1 ∂ ∂θ j log M i θ * 1≤i≤K, j∈m * . ( 7.33)
We also established a central limit theorem for θ(m * ) in (3.3), and this implies

∂ θ γ k (θ (k) ) 1≤k≤K √ n ( θ(m * )) i -θ * i i∈m * L -→ n→+∞ N K 0 , J K (m * ) F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 J K (m * ) . (7.34)
On the other hand, when θ = θ * , e 2 t (θ * ) = ξ 2 t for any t ∈ Z and since E[ξ 2 0 ] = 1, we deduce that e 2 t (θ * ) -1 t is a sequence of centred iid random variables with variance µ 4 -1 with µ 4 = E[ξ 4 0 ]. In such as case, the asymptotic behavior of the covariograms is well known and we deduce:

√ n γ k (θ * ) 1≤k≤K L -→ n→+∞ N K 0 , (µ 4 -1) 2 I K , (7.35)
with I k the (K × K) identity matrix.

We would like to use (7.31) for obtaining the asymptotic behavior of γ(m * ). In (7.34) Using the proof of the asymptotic normality of the QMLE (see for instance [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]), a Taylor expansion of log-likelihood for large n leads to

( θ(m * )) i -θ * i i∈m * ≈ -F (θ * , m * ) -1 1 n ∂ ∂θ L n (θ * ).
Therefore, the asymptotic cross expectation between ∂ θ γ k (θ

E ξ 2 i -1 ξ 2 j -1 ξ 2 j-k -1 ∂ ∂θ log M i θ * + 1 n n i=1 n j=k+1 E ξ i ξ 2 j -1 ξ 2 j-k -1 ∂ ∂θ f i θ * .
Using conditional expectations, we have E ξ 2 i -1 ξ 2 j -1 ξ 2 j-k -1 ∂ ∂θ log M i θ * = 0 for i = j since k ≥ 1. Moreover, for i = j, we obtain:

E ξ 2 i -1 ξ 2 j -1 ξ 2 j-k -1 ∂ ∂θ log M i θ * = (µ 4 -1) E ξ 2 i-k -1 ∂ ∂θ log M i θ * ,
which is the row k of matrix -(µ4-1) 2 J K (m * ). Similarly, and using the assumption E ξ 3 0 ] = 0, we obtain E ξ i ξ 2 j -1 ξ 2 j-k -1 ∂ ∂θ f i θ * = 0 for any i, j and k. As a consequence,

Cov √ n γ(θ * ) , ∂ θ γ k (θ (k) ) k √ n ( θ(m * )) i -θ * i i∈m * -→ n→∞ 1 2 (µ 4 -1) J K (m * ) F (θ * , m * ) -1 J K (m * ).
Finally, we deduce the asymptotic covariance matrix of √ n γ(m * ), which is

(µ 4 -1) 2 I K + J K (m * ) F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 J K (m * )
+ (µ 4 -1) J K (m * ) F (θ * , m * ) -1 J K (m * ).

Moreover the vector γ(m * ) is normal distributed from Lemma 3.3 of [START_REF] Ling | Diagnostic checking of nonlinear multivariate time series with multivariate arch errors[END_REF]. Thus, using Slutsky Lemma and with γ 0 (m * ) a.s.

-→ The proof is achieved after using the limit theorem (7.30).

imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

(2) (5.2) follows directly from (5.1).

(3) We follow a same reasoning like in the proof of Theorem 3. Applying the Total Probability Rule and by virtue of Theorem 3.1, we obtain:

F n (x) = P 1≤k≤K √ n ρ(m * ) k ≤ x k .
Therefore, the vectors √ n ρ( m) and √ n ρ(m * ) have exactly the same distribution.

Theorem 3 . 2 .

 32 Under the assumptions of Theorem 3.1 and if θ * ∈ o Θ and Var(Θ) hold, then

  imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

  consider the returns of the daily closing prices of the FTSE index of the London Stock Exchange 100. They are 2273 observations from January 4th, 2010 to December 31st, 2018. The mean and standard deviation of the returns are -0.54 and 57.67, respectively. The Time plot and the correlograns for the log returns and squared log returns are plotted in Figure 1. The Figures (1a) and (1c) exhibit the conditional heteroskedasticity in the log return time series. Moreover, Figure (1b) shows that more than 5 per cent of the autocorrelations are out of the confidence interval ±1.96/ √ 2273 and specially the Figure (1d) suggests that the strong white noise assumption cannot be sustained for this log-returns sequence of FTSE index.

  θ2 . imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019 (a) Time plot of log returns. (b) Correlograms of log returns. (c) Time plot of squared log returns. (d) Correlograms of squared log returns.

Figure 1 :

 1 Figure 1: Daily closing FTSE 100 index (January 4th, 2010 to December 31 st, 2018).

Lemma 3 .

 3 Under the assumptions of Theorem 3.1 and if a model m ∈ M is such that θ * ∈ Θ(m), then:1 κ n L n ( θ(m)) -L n (θ * ) = o P (1). (7.15)

Lemma 4 .

 4 Under the assumptions of Theorem 3.1 and for m ∈ M such as m * ⊂ m, then:L n ( θ(m)) -L n (θ * ) = O P (1). (7.21)

  Lemma 3 and because |m| -|m * | ≥ 1. 2. Let m ∈ M such as m * ⊂ m. Then,

From Lemma 4 ,

 4 one deduces L n ( θ(m * )) -L n (θ * ) = O P (1). In addition, in the sequel, we are going to show that L n (θ * ) -L n ( θ(m)) = n A(m) + o P (1) , with A(m) > 0. (7.23) For any θ ∈ Θ(m), we have from Proposition 2

3 . 1 ,

 31 P m = m * -→ n→∞ 1 and P m = m * -→ n→∞ 0. Therefore the second term in the right side of the previous equality asymptotically vanishes. For the first term, we can write,

n→+∞ µ 4 - 1 ,

 41 and with ρ k (m * ) = γ k (m * )/γ 0 (m * ), the limit theorem (5.1) holds withV (θ * , m * ) := I K + (µ 4 -1) -2 J K (m * ) F (θ * , m * ) -1 G(θ * , m * )F (θ * , m * ) -1 J K (m * ) + (µ 4 -1) -1 J K (m * ) F (θ * , m * ) -1 J K (m * ).(7.37)

2 .

 2 For x = (x k ) 1≤k≤K ∈ R K , denote by F n (x) = P 1≤k≤K √ n ρ( m) k ≤ x k the distribution function of √ n ρ( m).

  (for instance, the BIC penalization with κ n = log(n), or κ n = √ n) ensures the consistency of m and the Theorem (3.2) holds if in addition, θ * ∈

	o

1, imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

• if γ > 2, the condition κ n -→ n→∞ ∞

Table 1 :

 1 Percentage of selected order based on 1000 replications depending on sample's length for Model 1, 2 and 3 respectively.

		Sample length n Penalty	100 log n	√	500 n log n	√	1000 n log n	√	2000 n log n	√	n
		Wrong	21	32.3 3	0.9 0.9			0 0.2	0
	Model 1	True	74.6 67.5 95.8 99.1 98.2	100 99	100
		Overfitted	4.4		0.2 1.2		0 0.9			0 0.8	0
		Wrong	81.8 97.5 30.1 67.4 19.9 33.2 10.2 10.5
	Model 2	True	16.1		2.5 69.1 32.6 79.5 66.8 89.4 89.5
		Overfitted	2.1			0 0.8		0 0.6			0 0.4	0
		Wrong	78.9 92.9 25.7 70.5 11.6. 39.2 5.4	11.4
	Model 3	True	20.4		7.0 73.2 29.5 88.1 60.8 94.3 88.6
		Overfitted	0.1		0.1 1.1		0 0.3			0 0.3	0

Table 2 :

 2 The empirical size and empirical power of the portmanteau test statistic Q

	Sample length	100	500	1000	2000
		size power size power size power size power
		Model 1 3.5	13.6 3.8	48.1 3.5	82.7 3.2	97.7
	K = 3	Model 2 4.0	6.7 5	21.7 4.8	38.6 4.4	64.2
		Model 3 4.3	52.7 4.2	98.6 3.2	99.6 3.6	99.9
		Model 1 3.5	9.4 4.8	43.1 5.3	74.6 4.5	97.6
	K = 6	Model 2 2.1	6.3 4.9	18 4.5	32.2 6.4	61.3
		Model 3 3	18.3 3.1	91.5 3.4	99.6 6.8	99.7

K ( m) based on 1000 independent replications (in %) with K = 3 and K = 6.

Table 3 :

 3 Percentage of selected model based on 1000 replications depending on sample's length for Model 4

	Sample length	100 log n	√	500 n log n	√	1000 n log n √	2000 n log n √	n
	true model	85.9	68 97.5 100 96.8 100 98.9 100
	overfitted	7.9		2 2.5		0 3.2	0 1.1	0
	false model	6.2	30 0		0 0	0 0	0

Table 4 :

 4 Percentage of selected order based on 1000 replications depending on sample's length for model 5 log n n 2/3 log n n 2/3 log n n 2/3

	Sample length	100	500	1000	2000
	log n n 2/3 p < 2 8.9 69.1 0.1	17.6 0	6.1 0		0.9
	p = 2	88.9	30.9 75.3	82.4 71.4	93.9 72.5	99.1
	p > 2	2.2	0 24.6	0 28.6	0 27.5	0

  and 1

			√	n	∂Ln( θ(m)) ∂θ	= o P (1), it follows
	that:
	1 κ n	θ(m) -θ * ∂L n ( θ(m)) ∂θ	= o P (1).		(7.18)

  As we point out in Subsection 2.4, the proof is divided into two parts.Proof. 1. For m ∈ M such as m * ⊂ m and m = m * (overfitting), then using with C(m) = -2 L n θ(m) + |m| κ n (see (2.8)), we have:

	by using equality (7.19).
	7.2. Proof of Theorem 3.1
	1)	+	O P (1)
	= O P (1),
		imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

  Θ < ∞, we deduce from the same inequalities as in the first case of I 1 1 that I 2

					.26)
	b/ if X ⊂ AC( H θ ), same computations imply I 1 1 Θ	a.s. -→ n→+∞	0 when
	∞
	t -1/4	α j ( H θ , Θ) < ∞.	(7.27)
	t=1	j≥t
	Since E e 2
		1 Θ	a.s. -→ n→+∞	1 Θ 0 and I 3	a.s.
				1/2
					< ∞,	(7.28)

t (θ) Θ ≤ 2 h -1 E X 2 t + f t θ 2 Θ < ∞ and similarly E e 2 t (θ) -→ n→+∞ 0 when ∞ t=1 t -1/4 j≥t α j (f θ , Θ) + α j (M θ , Θ) + α j ( H θ , Θ) imsart-

ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

  and(7.35), we obtained the asymptotic normality of each of the two terms composing γ(m

* ). Now we need to study the joint asymptotic behavior of √ n γ(θ * ) and √ n ( θ(m * )) i -θ * i i∈m * . imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

  (k) ) k √ n ( θ(m * )) i -θ * i i∈m * and √ n γ(θ * ) is equal to: -J K (m * ) F (θ * , m * ) -1 E ∂ ∂θ L n (θ * ) γ(θ * ) .

												(7.36)
	From (2.5), a direct differentiation of L n provides
			∂ ∂θ	L n (θ * ) =	n t=1	e 2 t (θ * ) -1	∂ ∂θ	log M t θ * +	n t=1	e t (θ * )	∂ ∂θ	f t θ *
	so that,
	E	∂ ∂θ	L n (θ * ) γ k (θ * ) =	1 n	E	n i=1	e 2 i (θ * ) -1	∂ ∂θ	log M i θ *	n j=k+1	e 2 j (θ * ) -1 e 2 j-k (θ * ) -1
				+ j (θ = 1 n E n i=1 e i (θ * ) n ∂ ∂θ f i θ * j=k+1 e 2 1 n n n i=1 j=k+1

* ) -1 e 2 j-k (θ * ) -1

imsart-ejs ver. 2014/10/16 file: final_5.tex date: July 22, 2019

* Developed within the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 754362.

† Developed within the ANR BREAKRISK : ANR-17-CE26-0001-01.


Alireza Rezaei

Emanuel Aldea
email: emanuel.aldea@u-psud.fr

Piercarlo Dondi

Marco Malagodi

Sylvie Le Hégarat-Mascle

Detecting alterations in historical violins with optical monitoring

Keywords: A-contrario detection, UV fluorescence, historical violins, preventive conservation

Some historical musical instruments are still played today, and are prone to be affected by mechanical wear of superficial varnishes in direct contact with the musicians. In a previous work, an ad-hoc monitoring plan for historical violins, that involves the use multiple non-invasive analytic techniques, achieved good performance. However, the high number of techniques is a limitation if we want to perform frequent checks. In this work, our aim is to rely only on UV induced fluorescence images for performing a fast, preliminary check and then, if a possible alteration is detected, conduct spectroscopic analyses, which are more precise but also more time consuming. In this study, we explore the a-contrario framework in order to allow for the automatic detection of significant changes in the superficial varnishes. The difficulty of detecting the changes is represented by the need to define the significance of a change, in the absence of a ground truth provided by the expert. Tests performed on samples that simulate the effect of surface alteration during time show promising results.

INTRODUCTION

Preventive conservation is a field of research that focuses on reducing the risk of deterioration of artworks, with the final goal to minimize interventions on them. [START_REF] Bradley | Preventive conservation research and practice at the british museum[END_REF] Historical musical instruments are a very unique kind of artifacts, since they are both preserved in museums and still played today, leading to a risk of mechanical wear of superficial varnishes in direct contact with the musicians. A previous work [START_REF] Fichera | Innovative monitoring plan for the preventive conservation of historical musical instruments[END_REF] proposed an ad-hoc monitoring plan for historical instruments that involves the use of multiple non-invasive analytic techniques such as UV induced fluorescence (UVIFL or UVF) photography, colorimetric measures, XRF and FTIR spectroscopic analyses. In that study, two important violins were analyzed during a six months period: Vesuvio c.1727 (frequently played) made by Antonio Stradivari and Carlo IX c.1566 (rarely played) made by Andrea Amati. The achieved results were promising and allowed to monitor possible variations in the optical and chemical-physical properties of the varnishes. However, the high number of adopted techniques is a limitation if we intend to perform frequent checks on a larger set of instruments in order to detect the wear in the earliest possible stages. In this paper, we propose a new approach for searching the presence of variations in violins varnishes using only UVF images. UVF photography is a well-known non-invasive diagnostic technique, used in cultural heritage field, that allows to see details not perceivable using visible light. [START_REF] Janssens | Non-destructive micro analysis of cultural heritage materials[END_REF][START_REF] Dondi | Automatic analysis of uv-induced fluorescence imagery of historical violins[END_REF] This is due to the properties of different materials used in artworks that react to UV-A light (315 -400 nm) and re-emit radiations in the visible wavelength (400 -700 nm), producing characteristic fluorescence colors. [START_REF] Stuart | Analytical techniques in materials conservation[END_REF] Since the UV radiation only reaches the most superficial layers of a surface, UVF photography alone cannot be conclusive, but can give us some hints about the presence of color variations due to materials modifications. Our goal is to use UVF image processing as a preliminary fast examination of the surface of a violin and then, only if a possible alteration is detected, perform spectroscopic analyses (punctual examinations, more precise but also more time consuming, that can give information about chemical elements presence and organic and inorganic compounds 6 ). It is important to stress that this problem is different (and complementary) from monitoring the evolution of regions already worn out. [START_REF] Dondi | Automatic identification of varnish wear on historical instruments: The case of antonio stradivari violins[END_REF] In preventive conservation we want to detect the very early stages of an alteration even in areas where the varnishes are still present. Thus, we are interested in the detection of the smallest alterations that may occur on the surface of an instrument after a period of use.

The data input for our analysis is a change map for which the domain is a sub-region of the specimen (provided by an expert), and where change values are three-dimensional vectors defined in an adapted color space (i.e., HSV). The difficulty of detecting the changes is represented by the need to define the significance of a change, in the absence of a ground truth provided by the expert. Since it is not feasible to characterize directly the behavior of the target pixels (related to a possible alteration), the prior knowledge regarding the evolution of their appearance is very imprecise, and it is more effective to model the background data rather than to model accurately the category of interest.

In this study, we explored the a-contrario framework to allow for automatic detection of significant changes in the superficial varnishes. Since the seminal articles of Desolneux et al., [START_REF] Desolneux | Meaningful alignments[END_REF][START_REF] Desolneux | A grouping principle and four applications[END_REF] detection approaches based on the Number of False Alarms criterion (NFA) became more and more popular in the field of image processing. In these approaches, the notion 'a-contrario' refers to the fact that detection is performed by contradicting a 'naive' model that represents the null hypothesis (H 0 ) in statistical decision theory. By defining a measurement of deviation relatively to the naive model, the main asset of such approaches is their independence from threshold parameters, as they cast the detection as an optimization problem by maximizing the significance defined from deviation.

DATA PREPROCESSING

It is assumed that each sample in our series of violin surface photos has been acquired months apart from the others. Human errors in positioning the specimens lead to unavoidable differences in the appearance of the area of interest. Furthermore, aging equipment can produce different output from before. Therefore, it is necessary to account for these changes and to match as accurately as possible each time frame with the original image.

While the same illumination hardware setup and configuration are used during each capture, it is still possible for global or local differences to persist. To compensate the global variation problem, a robust approach which is effective as long as changes remain small, as in our case, is to equalize the average gray level pixel values of each frame, i.e. subtract the difference between average of two frames from the second frame:

Ī2 (p) = I 2 (p) -[µ(I 2 ) -µ(I 1 )]
Additionally, the other components of the setup -camera, sample holder and the sample itself -may have slight movements and introduce mismatches between frames. Therefore it is necessary to perform a spatial registration step as well.

Spatial registration

In order to spatially match the samples, we rely on extracting and matching SIFT 10 features in the original image and also in the subsequent frames. Many approaches exist for registering deformable objects performing general transformations, 11, 12 however our captures have been performed on a rigid object in a controlled environment. Although some residual rotations may subsist, the transformation for each image pair may be approximated using small translation and scaling components (the proper alignment of the samples with respect to the imager is easier to perform during the capturing process). Figure 1 shows the detected SIFT features in a sample pair of images and the result of registration. The figure also illustrates pairs of matched features which are, in our case, very tightly coupled. These matched points are used in the next step to estimate a proper transformation between the two frames using a robust estimation method. [START_REF] Torr | Mlesac: A new robust estimator with application to estimating image geometry[END_REF] After estimating the right transformation from the matched features and applying it to the moving image we assume that the alignment is achieved at pixel level. Although errors in registration may persist, this assumption is sufficient for the following process given the expected scale change.

Difference map computation

The previous assumption allows us to compute a difference map between each frame and the original (at time zero) frame. From now on, we will refer to the first frame as t 0 and other subsequent frames as t i .

The difference map between t 0 and t i is created by computing the color difference between the corresponding pixels. We have utilized the CIEDE2000 formula [START_REF] Luo | The development of the cie 2000 colour-difference formula: Ciede2000[END_REF] after transforming the RGB values into the CIELAB color space. Then, the resulting values have been normalized in order to fit into the [0, 1] range.

It is worth noting that other approaches may also be used to create the difference map for the rest of the detection process, from a simple euclidean distance between RGB values to more complicated color difference models. [START_REF] Robertson | The cie 1976 color-difference formulae[END_REF] However, in our case the employed difference provides consistent results which are in agreement with the expected wear areas.

WEAR DETECTION

In order to instantiate the a-contrario perception concept through a NFA criterion, two elements have to be defined: the 'naive' model that represents the statistics of the model to reject (the H 0 hypothesis) and the feature on which these statistics apply. Both depend on the considered data. However, since the naive model represents the absence of structure, we can choose it as representing the wide spreading of the samples, so that it will be rejected once the observations appear unlikely close with respect to the naive model.

Dealing with change detection, the decision of a change is due to the observation of a surprisingly high density of differences within local features. Such a definition can be interpreted as gathering two criteria: at pixel scale, high differences in feature images and, at region/area scale, high density of previously detected 'high differences'. In other words, we propose a two-step approach that firstly detects seeds as pixels likely to belong to a change area, and secondly detects dense areas of seeds. For each step, we use a NFA criterion to perform parameter-free decision.

Seed detection

Starting from the color difference image ∆I (Section 2.2) defined on the pixel domain P ⊂ N 2 , we consider the naive model M col to derive the set of seeds, called S ⊆ P, representing the pixels likely to belong to the altered areas. Specifically, denoting by |X| the cardinality of a set X, Definition 1 (Naive model M col ). The image ∆I is a random field of |P| independent centered Gaussian variables N 0, σ 2 .

According to M col , the distribution of the sum of the squared values (SSV) on a subdomain D ⊆ P,

υ D = s∈D [∆I(s)]
2 , is a χ 2 law with |D| degrees of freedom. Then, the probability P M col (υ D , σ) of observing a SSV lower than υ D by chance is given by the regularized incomplete Gamma function, and the Number of False Alarms associated to a subdomain D having υ D SSV is [START_REF] Robin | An a-contrario approach for subpixel change detection in satellite imagery[END_REF][START_REF] Aldea | Robust crack detection for unmanned aerial vehicles inspection in an a-contrario decision framework[END_REF]

N F A 1 (D, σ, |P|) = |P| |P| |D| P M col (υ D , σ) (1)
where a b is the binomial coefficient. Then, minimizing NFA defined by Eq. ( 1), the result depends on the parameter σ that controls the noise level in M col . In this case, similarly to prior work, [START_REF] Robin | An a-contrario approach for subpixel change detection in satellite imagery[END_REF][START_REF] Aldea | Robust crack detection for unmanned aerial vehicles inspection in an a-contrario decision framework[END_REF] it is computed by calculating the second moment of the image: [START_REF] Fichera | Innovative monitoring plan for the preventive conservation of historical musical instruments[END_REF] where µ is the statistical mean of the pixel values of the image ∆I.

σ 2 = E(x -µ)
Let us D = argmin D⊂P N F A 1 (D, σ, |P|). Since the naive model represents the inconsistency in the data, D is the set of pixels that are 'surprisingly' structured (with respect to the previous violin image) under the naive model assumption M col , i.e. the pixels presenting 'surprisingly' low ∆I values, so that the set of seeds S is the complementary of D with respect to set P: S = P \ D.

Wear area detection

Then, having derived S and represented it under the form of a binary image, we aim to detect the most significant cluster(s) of seeds. In this study, we compare two approaches: the first one assumes a parametric geometric shape of the altered areas (e.g. rectangular tiles, strips, rings, etc. like in Le Hégarat-Mascle et al. [START_REF] Hégarat-Mascle | Efficient evaluation of the number of false alarm criterion[END_REF] ), while the second approach considers a general shape clustering scheme proposed in Desolneux et al. [START_REF] Desolneux | A grouping principle and four applications[END_REF] In both cases, the considered naive model M bin represents the absence of spatially consistant subset(s) of seeds. Specifically, Definition 2 (Naive model M bin ). The set of seeds S is a random set of |S| independent uniformly distributed variables over the image lattice P.

Under uniform distribution model M bin , denoting by p O , the prior probability that a seed belongs to a parametric object O, the probability P M bin (p O , |S| , κ) of observing κ seeds within o by chance is given by the tail of the binomial distribution, and the Number of False Alarms 9 is

N F A 2 (p O , |S| , κ) = N test |S| i=κ |S| i p i O (1 -p O ) |S|-i (2)
In the previous equation, p O is estimated by the ratio between the area of object O with respect to the whole image area. Note the slight difference with a NFA like in [START_REF] Dibos | Real-time segmentation of moving objects in a video sequence by a contrario detection[END_REF] derived assuming a Bernoulli distribution of parameter p for pixel binary values, so that the probability to have a given number κ of seeds among a given number O of pixels is a Binomial distribution of parameter p and

N F A 2 (p, O, κ) = N test O i=κ O i p i (1 -p)
O-i , with p approximated by the ratio between the seed and the pixel numbers and O the pixel number of object O.

In the case of a clustering approach, instead of constraining the object in terms of parametric form, a thick low resolution curve free of any seed and surrounding the object is required. Thus, denoting by C a cluster, its relative area a(C) with respect to the whole image area is also the probability of a seed to belong to C under the naive model M bin , whereas the probability of a seed not to belong to C is 1 -a(C) -a(δC), where a(δC) is the relative area of the empty thick low resolution contour surrounding C. Transposing Desolneux' formula 9 with our notations,

N F A 2 (|C| , |S| , a(C), a(δC)) = M test |S| i=|C| |S| i [a(C)] i [1 -a(C) -a(δC)] |S|-i (3)
In Eq. ( 2) and ( 3), N test and M test are the numbers of tests that controls the average number of false alarms. [START_REF] Desolneux | From gestalt theory to image analysis: a probabilistic approach[END_REF] Conversely to the case of the first NFA (cf. Section 3.1), here, like in Desolneux et al., [START_REF] Desolneux | A grouping principle and four applications[END_REF] we take these numbers constant for a given image, i.e. independent of O of C, so that it is not involved in NFA minimization. Numerically, each cluster C is formed by traversing the minimum spanning tree created from the seed points S. Then using Eq. 3, for each cluster we compute the meaningfulness(equation 4). Finally, the wear areas are separate clusters with maximum meaningfulness.

S (|C| , |S| , a(C), a(δC)) = -log(N F A 2 (|C| , |S| , a(C), a(δC))) (4)
Figure 2 illustrates a simple synthetic example. A noisy binary image with a patch of higher density noise is given to the algorithm as an input. The higher density patch has been recognized as a meaningful cluster.

Implementation considerations

Assuming that each frame t i have around 250k pixels, computing and minimizing the first NFA (Eq. 1) is a demanding process. We have used special 20 digit numbers to have enough precision for this computation. Furthermore, for the minimization process it is more efficient to compute the log(N F A). Also, for the second NFA (Eq. 2), we have used the Hoeffding approximation 21 in order to compute the logarithm more efficiently:

-log(N F A 2 ((|C| , |S| , a(C), a(δC))))) ≈ |S| |C| |S| log |C| |S| * a(C) + 1 - |C| |S| log 1 -|C| / |S| 1 -a(C) -a(δC) (5)
In case of unacceptable performance problems, it is possible to resize the difference map to a more manageable size. In our case, an image of roughly 250k pixels gave us acceptable running time for the algorithm, more precisely approximately 20 seconds for the seed detection and 30 seconds for the clustering on non-optimized MATLAB implementations.

EXPERIMENTAL RESULTS

Construction of a gradual wear dataset

Since, at the moment, we have only few multi-temporal images of historical violins (further acquisitions will require several months to be significant), we created and artificially worn out a sample in laboratory for increasing our test dataset. First, the wooden sample was varnished by a violin maker in the same way adopted for a violin. Then, for simulating the effect of mechanical wear and of sweat (the most common sources of alteration), we scrubbed the sample surface using a cloth with a single drop of alcohol on it. The process was repeated twenty times to mimic the effect of a gradual alteration during a long period of time. At each step we took a UVF photo of the sample using the same protocol that we adopt for violins. [START_REF] Dondi | An interactive tool for speed up the analysis of UV images of Stradivari violins[END_REF] Only the left side of the sample was altered, maintaining the right side intact as reference. A series of UVF photos of the unaltered wood sample was taken before to start the alteration to record the initial conditions.

Wear detection using constrained shape primitives

Detection with shape primitives (in our case tiles and rings) has been performed on the 20 images of the altered sample when they are compared to the original unaltered surface. Figure 3 shows the result of the detection on four of these frames (t 1 ,t 6 ,t 9 and t 15 ), selected in order to highlight the performance on various levels of wear. In each case, significant clusters have been identified using rectangular tiles or circular disks and highlighted with a bounding rectangle or circle respectively. We have also applied the cluster detection process discussed in Sections 3.1 and 3.2 on all 20 images in the dataset. In each case, we have chosen four clusters based on the highest meaningfulness values (Eq. 4). Figure 4 shows the result of the algorithm for four of these frames: t 1 ,t 6 ,t 9 and t 15 . In each image, the clusters have been identified by a loose contour and the ranking of the cluster is shown by the number beside it. As it is evident, all major cluster have been correctly detected. Also, it is clear that the detection have completely ignored the noise and smaller artifacts appearing in the image.

As mentioned in Section 4.1, in both detection methods we are mainly interested in the left side of the sample, as it is the area where the change in varnish should appear. All four frames show signs of change in that area and as expected, this area keeps growing in subsequent frames. It is worth mentioning that although other relatively big clusters have been detected, they will not be considered as a worn out area since their area is constant across multiple frames. These change regions occur mainly due to the reflected UV light from the edges of the sample or unvarnished areas.

Discussion

The solution we propose aims to perform in a automated fashion a low-level detection, followed by a higher-level search for a pattern bearing evidence for the instrument wear. While the interest for the first step presented in Section 3.1 is clear in light of the very fine difference values observed as the wear progresses, for the second detection (Section 3.2) there is clearly a trade-off between the computational cost and the expressiveness of the pattern. Irrespective of the target configuration to be detected, we underline the fact that one of the advantages of this approach is to account for wear cues across a sub-domain of the image without relying necessarily on a strong connectivity prior of the seeds creating the pattern.

In terms of computational cost, the cluster detection process treats each seed as an independent point with respect to the others. Since in our application, the changed cluster will most likely be a dense grouping of points, it would be beneficial to consider each connected component as an individual node in the minimum spanning tree employed by the clustering algorithm. This way, we could decrease the number of clusters to check and benefit overall from a more efficient process.

CONCLUSION

In this work we studied the problem of temporal monitoring for the presence of alterations in the superficial varnishes of historical violins, using only UVF images. Our goal is to assess the feasibility of a quantitative solution for an early detection of alterations in the absence of ground truth data provided by experts. Extensive testing performed on samples that simulate the effect of surface alteration during time show that the detection process may be automated and that it is able to pinpoint changed areas from early stages of wear.

For future work, we intend to study avenues for improving the computational cost of the seed clustering. Besides, we aim to extend the seed detection process (Section 3.1) in order to assist as well with making a decision between various color difference approaches for creating the difference map. The same process can also help in finetuning the spatial and radiometric registrations.

Figure 1 :

 1 Figure 1: An example of spatial registration: (a) matched SIFT feature points; (b) difference of two frames before registration; (c) difference of two frames after registration. Brighter locations indicate higher differences.

  (a) A synthetic input example (b) Circular area detection (c) Rectangular area detection (d) Cluster detection

Figure 2 :

 2 Figure 2: A test run for change area detection: (a) a noisy, synthetic image used as input; (b) shape constrained change detection (circular area); (c) shape constrained change detection (rectangular area); (d) cluster density based area detection.

Figure 3 :

 3 Figure 3: Result of the tile and circular detection in each frame t i when compared to t 0 .

Figure 4 :

 4 Figure 4: Four most meaningful clusters in each frame t i when compared to t 0 .

ACKNOWLEDGMENTS

We would like to thank "Fondazione Museo del Violino Antonio Stradivari", "Friends of Stradivari", and "Distretto Culturale di Cremona" for their collaboration. This work was partially granted by "Fondazione Arvedi-Buschini" of Cremona.