Votre espace

Organisateur : P. Doukhan

**Groupe de travail dépendance**

** **

Institut Henri Poincaré

11 rue Pierre et Marie Curie

75231 Paris Cedex 05

Buts du GTD:

il s’agit de créer une dynamique autour des techniques liées à la dépendance de processus aléatoires. En effet au cours des dernières années, des méthodes nouvelles ont vu le jour. Il apparaît clairement que les participants de ce groupe de travail sont très représentatifs de ces techniques; une dynamique très forte de collaboration est attendue. On évoquera donc des modèles, leurs asymptotiques et on tentera de garder à l’esprit l’adéquation de ces modèles à des applications réelles. Certains exposés seront fondés sur des applications (en génomique, actuariat, géographie, agronomie, médecine, etc..) dans lesquelles ressort clairement la notion de dépendance en temps ou en espace. Des travaux transverses entre probabilités, statistiques et des disciplines appliquées sont l’objectif de ce groupe de travail dont la périodicité sera hebdomadaire ou bi-hebdomadaire selon les disponibilités de chacun

**Paul Doukhan**

**Université de Cergy Pontoise**

**Membre du laboratoire AGM UMR8088 et du Labex MME-DII**

Mardi 10, Salle 01 à 10h

**Jean Marc Bardet (Paris 1, SAMM) ****Title: Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process**

We first provides some key-results about the parametric and semi-parametric estimation of the Hurst parameter of long-memory processes. Then, we consider the particular case of the estimation of the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations.

Mardi 31, Salle 201 à 10h

*C. Y. Robert (ISFA) Cluster size distributions of extreme values for the Poisson-Voronoi tessellation.*

We consider the Voronoi tessellation based on a homogeneous Poisson point process in n R^d. For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson-Delaunay tessellation is also discussed.

http://arxiv.org/abs/1607.04075

Mardi 28, Salle 201 à 10h

*Oleg Klesov (Kiev) Repeated records*

We discuss some properties for sequential (repeated) records

constructed from sequences of dependent random variables.

Mardi 21, Salle 201 à 10h

*J. Collamore (Copenhague) *

**Title: Large deviations and conditioned limit laws for matrix recursive sequences**

Our main objective is to describe the large deviation behavior and limiting path properties for matrix recursive sequences. Motivated by branching processes in random environments, matrix recursions were originally introduced in the seminal paper of Kesten (1973), who studied the recursive sequence V(n) = A(n) V(n-1) + B(n), where {(A(n)} is an i.i.d. sequence of random matrices and {B(n)} an i.i.d. sequence of random vectors. Under a stationarity condition, Kesten showed that V(n) converges to a random variable V, whose tail decays at a specified polynomial rate. More recent work has applied this estimate in a variety of other areas, such as financial time series modeling.

In the first part of the talk, working under Kesten’s assumptions, we derive an extremal estimate for the first passage time and exhibit that, under a large excursion, the empirical average converges to an exponentially-shifted Markov random walk. Next, we turn to the case where the process {V(n)} is explosive, and derive a related large deviation estimate and certain conditioned limit theorems. (Based on collaborations with Sebastian Mentemeier and Anand Vidyashankar.)

Mardi 25 Avril, Salle 314 à 10h

**Konstantinos Fokianos (Cyprus) **

**Title: Testing independence for multivariate time series by the auto-distance correlation matrix**

We introduce the notions of multivariate auto-distance covariance and correlation functions for time series analysis.

These concepts have been recently discussed in the context of both independent and dependent data but we extend them in a different direction by putting forward their matrix version.

Their matrix version allows us to identify possible interrelationships among the components of a multivariate time series. Interpretation and consistent estimators of these new concepts are discussed. Additionally, we develop a test for testing the hypothesis of i.i.d. for multivariate time series data. The resulting test statistic performs better than the standard multivariate Ljung-Box test statistic. This is joint work with M. Pitsillou.

Mardi 15, Salle 421 à 10h

*Adam Jakubowski (Torun) *

*Stable limits for Markov chains**via Principle of Conditioning**Adam Jakubowski**Nicolaus Copernicus University, Toru_n, Poland**adjakubo @ mat.umk.pl*

Abstract: The Principle of Conditioning is a heuristic rule that allows transferring

limit theorems for independent random variables into limit theorems for depen-

dent random variables. While the limit theorems obtained via the Principle of

Conditioning are commonly known, methods of verifying their assumptions are

always of interest due to potential applications.

Our motivation comes from the work by Jara, Komorowski and Olla (2009),

where the functional limit theorems due to Durret and Resnick (1978) were

used to obtain a fractional di_usion as a scaled limit of solutions of a linear

Boltzmann equation (with Markov chains providing probabilistic solutions of

the latter).

We have obtained a di_erent set of conditions ensuring that partial sums of

a stationary Markov chain converge to a stable law with exponent _ 2 (0; 2).

The conditions are related to operator properties of the transition probabilities.

The approach is through the \Main Lemma" of the Principle of Conditioning.

This is a joint work with Mohamed El Machkouri and Dalibor Voln_

Mardi 22, Salle 314 à 10h

*Ivan Nourdin "Phase Singularities in complex arithmetic Random Waves"*

** r**ésumé: "Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We will use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result will be that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. The talk will be based on a joint work with Federico Dalmao, Giovanni Peccati and Maurizia Rossi. »

Mardi 29, Salle 201 à 10h

*Aboubacar Amiri (Lille) "Regression estimation by local polynomial fitting for multivariate data streams"*

Abstract: In this paper we study a local polynomial estimator of the regression function and its derivatives. We propose a sequential technique based on a multivariate counterpart of the stochastic approximation method for successive experiments for the local polynomial estimation problem. We present our results in a more general context by considering the weakly dependent sequence of stream data, for which we provide an asymptotic bias-variance decomposition of the considered estimator. Additionally, we study the asymptotic normality of the estimator and we provide algorithms for the practical use of the method in data streams framework.

Mardi 4, Salle 201 à 10h

*TBA*

Mardi 18, salle 201 à 10h

*Jean Michel Zakoian (ENSAE)*

*Conditional VaR estimation for dynamic portfolios driven by multivariate GARCH models*

* Abstract*

We study the estimation risk induced by univariate and multivariate methods for evaluating the conditional Value-at-Risk (VaR) of a portfolio

of assets. The composition of the portfolio can be time-varying and the individual returns are assumed to follow a general multivariate

dynamic model. Under ellipticity of the conditional distribution, we introduce in the multivariate framework a concept of VaR parameter,

and we establish the asymptotic distribution of its estimator. A multivariate Filtered Historical Simulation method, which does not rely on ellipticity, is studied.

We also consider two univariate approaches based on past real or reconstituted returns. We derive asymptotic confidence intervals for the conditional VaR, which allow to quantify simultaneously the market and estimation risks. Potential usefulness, feasibility and drawbacks of the different univariate and multivariate approaches are illustrated via Monte-Carlo experiments and an empirical study based on stock returns.

** **

Mardi 6, Salle 201 à 10h

**Paul Doukhan (Cergy-Pontoise).**

**Projets, planification et organisation du GTD.**

La première séance du GTD aura pour objet de mettre en place son organisation; je me permettrais aussi de lister quelques projets en cours pour ouvrir des discussions contradictoires,

- chaluts discrets,

- un nouveau point de vue sur la non stationnarité,

- sélection de modèle et identifiabilité,

- modélisation de tables de mortalité,

- estimation de la densité des innovations de modèles ARCH,

- modèles de séries temporelles de marginales Bernouilli.

Mardi 13, Salle 201 à 10h

* Quentin Gilbert (ISFA Lyon) Pricing and Risk Analysis of a Long-Term Care Insurance Contract in a non-Markov Multi-State Model*

Multi-state models are generally used for pricing and reserving long-term care (LTC) insurance contracts. While most of the current researches assume that the model is Markovian, we show in this paper that this assumption should actually be rejected, as it leads to a bias in the estimation procedure that may be significant. Since the transition probabilities are complex to estimate with an inhomogeneous semi-Markov model based on transition intensities, we choose to apply recent methods for a direct estimation of transition probabilities, which perform better than the Aalen-Johansen estimator when the Markov assumption is not satisfied. Using the so-called pseudo-values related to Jackknife methods on these estimators, we incorporate the effects of covariates (duration, sex and generation) with a GLM regression model, similarly to Helms et al. [Helms, F., Czado, C. and Gschlößl, S. Calculation of Premiums LTC based on Direct Estimates of transition probabilities. Astin Bulletin, 2005, 455-469]. Another key interest of our approach is to include the diseases which cause the entry into dependency as it affects strongly the residual lifetime of LTC claimants. We apply it on a real French LTC insurance portfolio and analyze the effect of Markov hypothesis both on pricing and on the solvency capital requirement calculation.

Mardi 27, Salle 201 à 10h

*I. Grublite (Cergy & Vilnius) TBA*

Mardi 21, salle 05 à 10h

*Yahia Sahli (ISFA, Lyon) LARCH-Type Approach for Modeling Mortality Improvements Surface*

Abstract: In this paper we consider the projection of mortality surface at the national level. We consider modeling mortality improvements on a cohort basis taking into account correlations across generations. Therefore, we propose to model the whole mortality surface by considering a random field approach with a specific causal structure instead of a univariate modeling framework. Such an approach has the advantage to account for a local dependence among adjacent cohorts.

Mardi 31, salle 421 à 10h

*Mathieu Rosenbaum (Paris 6) Nearly unstable Hawkes processes.*

Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L^{1} norm of their kernel is close to unity. We study in this talk such processes for which the stability condition is almost violated, focusing in particular on limit theorems.

Mardi 24, salle 201 à 15h

*Jacek Leśkow (Cracow University of Technology) A class of nonstationary, periodically or almost periodically correlated time series that are weakly dependent.*

Abstract: The focus will be on applications in telecommunication and mechanical signal processing. In a relatively simple model we will show how to study the asymptotic properties of the estimators. We will also introduce the concept of resampling and investigate some aspects of consistency of selected resampling procedures.

Mardi 3, salle 05 à 10h

*Deny Pommeret (Marseille) Data driven smooth test of comparison for dependent*

*sequences *

Abstract: We present a smooth test of comparison for the marginal distributions of two dependent stationary sequences.

We describe the general test procedure. Several situations of dependence are then investigated. We also indicate various perspectives. (joint work with P. Doukhan and L. Reboul)

Mardi 26, salle 201 à 10h

*Joseph Rynkiewicz (Paris 1) Asymptotic properties of autoregressive regime-switching models.*

Abstract: The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for Gaussian mixtures. We study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its asymptotic distribution. Some numerical examples illustrate the results and their convergence properties.