Paul Doukhan

Paul Doukhan
CYU

PhD
Looking for a postdoc in data sciences https://doukhan.u-cergy.fr/pdf/Postdoc2022.pdf

About

200
Publications
28,080
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4,885
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Introduction
Advisor for 20 PhD/Habilitation theses, I am involved into dependence structures. I introduced weak dependence with Sana Louhichi beyond mixing, to model real data. It includes incomplete data, integer valued models and high dimensional data. Features of the real data are both their non stationarity, their dimension and the way they are sampled. We also deal with applications to astronomy, insurance. Ecology is a main societal question whose approach necessitates the above techniques.
Additional affiliations
October 2002 - July 2008
September 2002 - March 2020
Université de Paris 1 Panthéon-Sorbonne
Position
  • Professor (Full)
September 1993 - present
Université de Cergy-Pontoise
Position
  • Professor (Full)
Education
July 1981 - May 1986
Université Paris-Sud 11
Field of study
  • Statistics
September 1979 - June 1981
Université Paris-Sud 11
Field of study
  • Statistics
October 1975 - September 2020

Publications

Publications (200)
Article
This paper extends the study of kernel-based estimation for locally stationary processes proposed in Dahlhaus et al., 2019 to infinite-memory processes models such as locally stationary AR(∞), GARCH(p,q), ARCH(∞) or LARCH(∞) processes. The estimators are computed as localized M-estimators for every contrast satisfying appropriate regularity conditi...
Article
Full-text available
This article proposes an optimal and robust methodology for model selection. The model of interest is a parsimonious alternative framework for modeling the stochastic dynamics of mortality improvement rates introduced recently in the literature. The approach models mortality improvements using a random field specification with a given causal struct...
Preprint
Full-text available
Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cram\'{e}r moderate deviation results for the Lotka-Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.
Preprint
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for Statistical Applications sake
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We introduce a class of Markov chains, that contains the model of stochastic approximation by averaging and non-averaging. Using martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differenc...
Article
Full-text available
We consider models for count variables with a GARCH-type structure. Such a process consists of an integer-valued component and a volatility process. Us- ing arguments for contractive Markov chains we prove that this bivariate process has a unique stationary regime. Furthermore, we show absolute regularity (β-mixing) with geometrically decaying coef...
Preprint
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We derive mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we prove absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. We provide easily verifiable sufficient conditions fo...
Preprint
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Taylor's power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. It has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics and mathematics. In particular, i...
Article
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We study nonlinear infinite order Markov switching integer‐valued ARCH models for count time series data. Markov switching models take into account complex dynamics and can deal with several stylistic facts of count data including proper modelling of nonlinearities, overdispersion and outliers. We study structural properties of those models. Under...
Preprint
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The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive latent process which forms a Markov chain in random environments. Contrarily to existing contributions in the f...
Article
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We propose a vector auto-regressive model with a low-rank constraint on the transition matrix. This model is well suited to predict high-dimensional series that are highly correlated, or that are driven by a small number of hidden factors. While our model has formal similarities with factor models, its structure is more a way to reduce the dimensio...
Preprint
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We consider integer-valued GARCH processes, where the count variable conditioned on past values of the count and state variables follows a so-called Skellam distribution. Using arguments for contractive Markov chains we prove that the process has a unique stationary regime. Furthermore, we show asymptotic regularity ($\beta$-mixing) with geometrica...
Preprint
Full-text available
This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of local stationary processes. We extend the results of Dahlhaus et al. (2018) considering the local stationary version of the infinite memory processes of Doukhan and Wintenberger (2008). The estimators are computed as l...
Article
We are studying linear and log-linear models for multivariate count time series data with Poisson marginals. For study- ing the properties of such processes we develop a novel conceptual framework which is based on copulas. Earlier contributions impose the copula on the joint distribution of the vector of counts by employing a continuous exten- sio...
Preprint
Full-text available
This article proposes an optimal and robust methodology for model selection. The model of interest is a parsimonious alternative framework for modeling the stochastic dynamics of mortality improvement rates introduced by Doukhan et al. (2017). The approach models mortality improvements using a random field specification with a given causal structur...
Preprint
Full-text available
Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (a j) j$\in$N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process ($\gamma$(t)) t$\in$R called the seed process. They provide a general framework for modeling linear or non-linear...
Chapter
The existing literature on extremal types theorems for stationary random processes and fields is, until now, developed under either mixing or “Coordinatewise (Cw)-mixing” conditions. However, these mixing conditions are very restrictives and difficult to verify in general for many models. Due to these limitations, we extend the existing theory, con...
Article
Full-text available
Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (aj)j∈N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process (γ(t))t∈R called the seed process. They provide a general framework for modeling linear or non-linear long range depende...
Article
Full-text available
In this paper, we adapt a data-driven smooth test to the comparison of the marginal distributions of two independent, short or long memory, strictly stationary linear sequences. Some illustrations are shown to evaluate the performances of our test.
Article
In this paper, we adapt a data-driven smooth test to the comparison of the marginal distributions of two independent, short or long memory, strictly stationary linear sequences. Some illustrations are shown to evaluate the performances of our test.
Preprint
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We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.
Article
Full-text available
We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.
Article
Full-text available
Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almo...
Preprint
Full-text available
We propose a vector auto-regressive (VAR) model with a low-rank constraint on the transition matrix. This new model is well suited to predict high-dimensional series that are highly correlated, or that are driven by a small number of hidden factors. We study estimation, prediction, and rank selection for this model in a very general setting. Our me...
Preprint
Full-text available
Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almo...
Preprint
Full-text available
We study nonlinear mixtures of integer-valued ARCH type models for count time series data. We investigate the theoretical properties of these processes and we prove ergodicity and stationarity, under minimal assumptions. The model can be generalized by including a GARCH component but we show that such inclusion can be accommodated by an ARCH model...
Chapter
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Research Proposal
Full-text available
This is a research project in Valparaiso for modelling astronomical data through non stationary models
Article
We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent ‘seed’ processes on shrinking intervals (‘trawl heights’). Related trawl processes in continuous time were studied in Barndorff-Nielsen et al. (2011, 2014). In the case when the trawl function decays as a...
Chapter
This chapter describes a simple Gaussian limit theory; namely we restate simple central limit theorems together with applications and moment/exponential inequalities for partial sums behaving asymptotically as Gaussian random variables. A relevant reference for the whole chapter is Petrov (Limit theorems of probability theory. Sequences of independ...
Chapter
We propose an overview of the notions of dependence in this chapter, good references are Doukhan et al. (Theory and applications of long-range dependence. Birkhaüser, Boston, 2002b) for long-range dependence, and Doukhan (Mixing: properties and examples. Lecture notes in statistics. Springer, New York, 1994) and Dedecker et al. (Weak dependence: wi...
Chapter
The notion of association, or positive correlation, was naturally introduced in two different fields: reliability (Esary, Proschan, Walkup in Annal Math Stat 38:1466–1474, 1967) and statistical physics (Fortuin, Kasteleyn, Ginibre in Commun Math Phys 22–2:89–103, 1971) to model a tendency that the coordinates of a vector valued random variable admi...
Chapter
We consider stationary sequences generated through independent identically distributed \((\xi _n)_{n\in \mathbb {Z}}\). A reference is Brockwell and Davis (Time series: theory and methods. Springer, New York, 1991). Such models are natural in signal theory since they appear through linear filtering of a white noise. The usual setting is that \((\xi...
Chapter
This chapter aims at describing stationary sequences generated from independent identically distributed samples \((\xi _n)_{n\in \mathbb {Z}}\). Most of the material in this chapter is specific to this monograph so that we do not provide a global reference. However Rosenblatt (Stationary processes and random fields. Birkhäuser, Boston, 1985) perfor...
Chapter
The present chapter deals with the standard notion of stochastic independence. This is a crucial concept, since this monograph aims to understand how to weaken it, in order to define asymptotic independence. We discuss in detail the limits of this idea through various examples and counter-examples.
Chapter
This chapter is devoted to moment methods. The use of moments relies on their importance in deriving asymptotic of several estimators, based on moments and limit distributions. Cumulants are linked with spectral or multispectral estimation which are main tools of time series analysis. $$g(\lambda )=\sum _{k=-\infty }^\infty \mathrm {Cov}\,(X_0,X_k)...
Chapter
Some bases for the theory of time series are given below. The chapter deals with the widely used assumption of stationarity which yields a simpler theory for time series. This concept is widely considered in Rosenblatt (Stationary processes and random fields, Birkhäuser, Boston, 1985) and in Brockwell and Davis (Time series: theory and methods, 2nd...
Chapter
Gaussian distributions (Appendix A) are natural and play a special role in the field of probability theory since they appear as limit distributions from the CLT (Theorem 2.1.1, Lemma 11.5.1). Gaussian linear spaces admit a simple geometric property.
Chapter
This chapter introduces some simple ideas. We investigate conditions on time series such that the standard limit theorems obtained for independent identically distributed sequences still hold. After a general introduction to weak-dependence conditions an example states the fact that the most classical strong-mixing condition from (Rosenblatt (1956)...
Chapter
Long-range dependent (LRD) phenomena were first exhibited by Hurst for hydrology purposes. This phenomenon occurs from the superposition of independent sources, e.g. confluent rivers provide this behaviour (see Fig. 4.2). Such aggregation procedures provide this new phenomenon. Hurst (Trans Am Soc Civ Eng 116:770–799, 1951) originally determined th...
Chapter
Many statistical procedures are derived from probabilistic inequalities and results; such procedures may need more precise bounds as this is proved in the present chapter for the independent case. Basic notations are those from Appendix B.1. Developments may be found in van der Vaart (Asymptotic statistics, Cambridge University Press, Cambridge, 19...
Book
https://www.springer.com/fr/book/9783319769370 This book presents essential tools for modelling non-linear time series. The first part of the book describes the main standard tools of probability and statistics that directly apply to the time series context to obtain a wide range of modelling possibilities. Functional estimation and bootstrap are...
Article
The existing literature on extremal types theorems for stationary random processes and fields are, until now, developed under either mixing or “Coordinate- wise (Cw)-mixing” conditions. However, these mixing conditions are very restric- tives and difficult to verify in general. In this context, we provide an extremal types theorem for stationary ra...
Article
In this article we propose a quasi-Whittle estimator for parametric families of time series models in the presence of missing data. This estimator extends results to the incompletely observed case. This extension is valid to non-Gaussian and nonlinear models. It also allows us to bound the variance of an associated quasiperi-odogram. A simulation s...
Book
This book presents essential tools for modelling non-linear time series. The first part of the book describes the main standard tools of probability and statistics that directly apply to the time series context to obtain a wide range of modelling possibilities. Functional estimation and bootstrap are discussed, and stationarity is reviewed. The sec...
Article
Full-text available
We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p,q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This...
Article
Full-text available
This article proposes a parsimonious alternative approach for modeling the stochastic dynamics of mortality rates. Instead of the commonly used factor-based decomposition framework, we consider modeling mortality improvements using a random field specification with a given causal structure. Such a class of models introduces dependencies among adjac...
Article
Full-text available
Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes (Xt), with local stationarity and periodic features (with a known period T), inducing the definition Xt = at(t/nT)X t--1 + $\xi$t for t $\in$ N and with a t+T $\not\equiv$ at. Central limit theorems are es...
Article
Full-text available
We are studying the problems of modeling and inference for multivariate count time series data with Poisson marginals. The focus is on linear and log-linear models. For studying the properties of such processes we develop a novel conceptual framework which is based on copulas. However, our approach does not impose the copula on a vector of counts;...
Cover Page
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Chapter
With the simple example of an asymmetric ARCH(1) model as a pretext, we introduce some of the main tools for weak dependence conditions introduced in [7]. The power of weak dependence is shown up for this very elementary model. This a special case of the infinite memory models in [8]. Asymptotic properties of a moment based parametric estimation do...
Article
Gene promoters have variable repartition of AGCT nucleotides according to some probabilistic behaviours essentially depending on their position in a string. The paper aims to provide a model for this configuration. With this model we derive non-uniform confidence bounds for those probability distributions in the strings. A uniform bound deriving fr...
Article
Full-text available
We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent `seed' processes on shrinking intervals (`trawl heights'). Related trawl processes in continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen et al. (2014), however in our case, the...
Article
We propose a general definition for weak dependence of point processes as an alternative to mixing definitions. We give examples of such weak dependent point processes for the families of Neyman Scott processes or Cox processes. For these processes, we consider the empirical estimator of the empty space function . Using the general setting of the w...
Article
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We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it can be written as a func...
Article
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The notion of a phantom distribution function (phdf) was introduced by O'Brien (1987). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any $\alpha$-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationa...
Article
Understanding the way extreme values do cluster in the case of time series is an essential problem for extreme values theory and risk management. Drees and Rootzen (2010) provided a deep solution to this problem. Anyway the existing literature on functional central limit theorems (FCLT) for empirical processes of cluster functionals (EPCF) is, unti...
Article
In this paper, we propose a model-free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so-called dependent wild bootstrap, which was introduced by Shao (2010) and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We s...
Chapter
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We study the almost sure limiting behavior of record times and the number of records, respectively, in a (so-called) \(F^\alpha \)-scheme. It turns out that there are certain “dualities” between the latter results, that is, under rather general conditions strong laws for record times can be derived from the corresponding ones for the number of reco...
Data
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Article
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In this paper we propose a smooth test of comparison for the marginal distributions of strictly stationary dependent bivariate sequences. We first state a general test procedure and several cases of dependence are then investigated. The test is applied to both simulated data and real datasets.
Article
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We discuss a class of conditionally heteroscedastic time series models satisfying the equation $r_t= \zeta_t \sigma_t$, where $\zeta_t$ are standardized i.i.d. r.v.'s and the conditional standard deviation $\sigma_t$ is a nonlinear function $Q$ of inhomogeneous linear combination of past values $r_s, s<t$ with coefficients $b_j$. The existence of s...
Article
We consider generalized linear models for regression modeling of count time series. We give easily verifiable conditions for obtaining weak dependence for such models. These results enable the development of maximum likelihood inference under minimal conditions. Some examples which are useful to applications are discussed in detail.
Article
Full-text available
We determine the almost sure and central limiting behaviour of the number of records and record times for the F α -scheme. Elementary methods are used to obtain general results. The basic results are extended to a random environments model.
Article
Full-text available
We consider here together the inference questions and the change-point problem in Poisson autoregressions (see Tj{\o}stheim, 2012). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it is shown that the conditional mean can be written...
Article
We investigate the relationship between weak dependence and mixing for discrete valued processes. We show that weak dependence implies mixing conditions under natural assumptions. The results specialize to the case of Markov processes. Several examples of integer valued processes are discussed and their weak dependence properties are investigated b...
Article
This paper deals with a general class of observation-driven time series models with a special focus on time series of counts. We provide conditions under which there exist strict-sense stationary and ergodic versions of such processes. The consistency of the maximum likelihood estimators is then derived for well- specified and misspecified models.
Article
We consider generalized linear models for regression modeling of count time series. We give easily verifiable conditions for obtaining weak dependence for such models. These results enable the development of maximum likelihood inference under minimal conditions. Some examples which are useful to applications are discussed in detail.
Article
Full-text available
We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety of examples, including mixing processes of different kinds. We present some appl...
Article
Full-text available
The aim of this paper is to provide a comprehensive introduction for the study of ℓ1-penalized estimators in the context of dependent observations. We define a general ℓ1-penalized estimator for solving problems of stochastic optimization. This estimator turns out to be the LASSO [Tib96] in the regression estimation setting. Powerful theoretical gu...
Article
Full-text available
The aim of this paper is to provide a comprehensive introduction for the study of L1-penalized estimators in the context of dependent observations. We define a general $\ell_{1}$-penalized estimator for solving problems of stochastic optimization. This estimator turns out to be the LASSO in the regression estimation setting. Powerful theoretical gu...
Article
Full-text available
Fazekas and Klesov (2000) found conditions for almost sure con-vergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of Móricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law o...
Article
Full-text available
This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series...
Article
Full-text available
This paper provides extensions of the work on subsampling by Bertail et al.in J. Econ. 120:295–326 (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi in Stoch. Proc. Appl. 84:313–342 (1999). We investigate properties of smooth and rough subsampling estimators for sampling distributions of...
Article
Full-text available
Self-normalized central limit theorems are important for statistical pur-poses. A simple way to achieve them is to consider estimations of the limit variance; this expression writes as a complicated covariance series under weak dependence. Using an argument of Carlstein (1986), we work out this program for a new procedure, in the case of vector val...
Article
Full-text available
Fazekas and Klesov (2000) found conditions for almost sure convergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of M´oricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law o...
Article
Full-text available
This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series...