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Jérôme Dedecker

Publications and source records attributed to Jérôme Dedecker.

At least 19 recordsLinked to original sources
arXiv2026

Choosing the penalty in nonparametric regression: short and long-range dependence

In this work, we study the one-dimensional regression problem under random design and Gaussian errors. Our framework is very general: we make no prior assumptions about the design (which may be nonstationary and exhibit short or long-range dependence), nor do we assume that the errors are homoscedastic. We examine in detail the cases where the error process exhibits short or long-range dependence. We adopt a least-squares penalized strategy using piecewise polynomials to estimate the regression function, following the framework of Baron, Birg{é} and Massart [1999]. We derive explicit penalties, up to calibration constants, to obtain adaptive estimators for which we establish risk bounds. Since these penalties depend on the dependence properties of the error process, which are unknown in practice, we propose several adaptations of the dimension jump calibration algorithm to make our procedures fully data-driven.

math.ST↗
arXiv2026

Normal approximation for partial sums: general convex costs

We provide non-asymptotic bounds and asymptotic limits for convex transport costs between the distribution of partial sums of independent and identically distributed square integrable and centered random variables and the normal distribution with mean zero and the same variance. The proof relies on controlling the transport cost by an appropriate ideal distance, combined with an adaptation of Lindeberg's method. The numerical constants and the asymptotic constants are explicit.

math.PR↗
arXiv2026

Concentration of the empirical measure in Wasserstein distance: bounds involving the covering dimension

We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete extension of the concentration inequalities of Fournier and Guillin [2015] in the case where E = R^d , in which the covering dimension replaces the dimension of the ambient space E.

math.PR↗
arXiv2025

Z-scores-based methods and their application to biological monitoring: An extended analysis of professional soccer players and cyclists athletes

The increase in the collection of biological data allows for the individual and longitudinal monitoring of hematological or urine biomarkers. However, identifying abnormal behavior in these biological sequences is not trivial. Moreover, the complexity of the biological data (correlation between biomarkers, seasonal effects, etc.) is also an issue. Z-score methods can help assess the abnormality in these longitudinal sequences while capturing some features of the biological complexity. This work details a statistical framework for handling biological sequences using three custom Z-score methods in the intra-individual variability scope. These methods can detect abnormal samples in the longitudinal sequences with respect to the seasonality, chronological time or correlation between biomarkers. One of these methods is an extension of one custom Z-score method to the Gaussian linear model, which allows for including additional variables in the model design. We illustrate the use of the framework on the longitudinal data of 3,936 professional soccer players (5 biomarkers) and 1,683 amateur or professional cyclists (10 biomarkers). The results show that a particular Z-score method, designed to detect a change in a series of consecutive observations, measured a high proportion of abnormal values (more than three times the false positive rate) in the ferritin and IGF1 biomarkers for both data sets. The proposed framework and methods could be applied in other contexts, such as the clinical patient follow-up in monitoring abnormal values of biological markers. The methods are flexible enough to include more complicated biological features, which can be directly incorporated into the model design.

stat.AP↗
arXiv2025

DEViaN-LM: An R Package for Detecting Abnormal Values in the Gaussian Linear Model

The DEViaN-LM is a R package that allows to detect the values poorly explained by a Gaussian linear model. The procedure is based on the maximum of the absolute value of the studentized residuals, which is a free statistic of the parameters of the model. This approach makes it possible to generalize several procedures used to detect abnormal values during longitudinal monitoring of certain biological markers. In this article, we describe the method used, and we show how to implement it on different real datasets.

cs.MS↗
arXiv2025

On the weak invariance principle for random fields with commuting filtrations under L1-projective criteria

We consider a field $f \circ T_1^{i_1} \circ \cdots \circ T_d^{i_d}$ where $T_1, \dots , T_d$ arecommuting transformations, one of them at least being ergodic. Considering the case of commuting filtrations, we are interested by giving sufficient ${\mathbb L}^1$-projective conditions ensuring that the normalized partial sums indexed by quadrants converge in distribution to a normal random variable. We also give sufficient conditions ensuring the weak invariance principle for the partial sums process. For the central limit theorem (CLT), the proof combines a truncated orthomartingale approximation with the CLT for orthomartingales due to Voln{ý}. For the functional form, a new maximal inequality is needed and is obtained via truncation techniques, blocking arguments and orthomartingale approximations. The case of completely commuting transformations in the sense of Gordin can be handled in a similar way. Application to bounded Lipschitz functions of linear fields whose innovations have moments of a logarithmic order will be provided, as well as an application to completely commuting endomorphisms of the m-torus. In the latter case, the conditions can be expressed in terms of the ${\mathbb L}^1$-modulus of continuity of $f$.

math.PR↗
arXiv2024

The robusTest package: two-sample tests revisited

The R package robusTest offers corrected versions of several common tests in bivariate statistics. We point out the limitations of these tests in their classical versions, some of which are well known such as robustness or calibration problems, and provide simple alternatives that can be easily used instead. The classical tests and theirs robust alternatives are compared through a small simulation study. The latter emphasizes the superiority of robust versions of the test of interest. Finally, an illustration of correlation's tests on a real data set is also provided.

stat.ME↗
arXiv2023

Transport collapse in dynamically evolving networks

Transport in complex networks can describe a variety of natural and human-engineered processes including biological, societal and technological ones. However, how the properties of the source and drain nodes can affect transport subject to random failures, attacks or maintenance optimization in the network remain unknown. In this paper, the effects of both the distance between the source and drain nodes and of the degree of the source node on the time of transport collapse are studied in scale-free and lattice-based transport networks. These effects are numerically evaluated for two strategies, which employ either transport-based or random link removal. Scale-free networks with small distances are found to result in larger times of collapse. In lattice-based networks, both the dimension and boundary conditions are shown to have a major effect on the time of collapse. We also show that adding a direct link between the source and the drain increases the robustness of scale-free networks when subject to random link removals. Interestingly, the distribution of the times of collapse is then similar to the one of lattice-based networks.

cond-mat.stat-mech↗
arXiv2023

Adaptive directional estimator of the density in R^d for independent and mixing sequences

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, $α$-mixing and $τ$-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. In particular, if A is an invertible matrix, if the observations are drawn from X $\in$ R^d , d $\ge$ 1, it achieves the rate implied by the regularity of AX, which may be more regular than X. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.

math.ST↗
arXiv2023

Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

math.PR↗
arXiv2021

Rates of convergence in the central limit theorem for martingales in the non stationary setting

In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${\mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/\sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {\rm Var}(S_n)$. Applications to linear statistics, non stationary $ρ$-mixing sequences and sequential dynamical systems are given.

math.PR↗
arXiv2020

Gaussian linear model selection in a dependent context

In this paper, we study the nonparametric linear model, when the error process is a dependent Gaussian process. We focus on the estimation of the mean vector via a model selection approach. We first give the general theoretical form of the penalty function, ensuring that the penalized estimator among a collection of models satisfies an oracle inequality. Then we derive a penalty shape involving the spectral radius of the covariance matrix of the errors, which can be chosen proportional to the dimension when the error process is stationary and short range dependent. However, this penalty can be too rough in some cases, in particular when the error process is long range dependent. In a second part, we focus on the fixed-design regression model assuming that the error process is a stationary Gaussian process. We propose a model selection procedure in order to estimate the mean function via piecewise polynomials on a regular partition, when the error process is either short range dependent, long range dependent or anti-persistent. We present different kinds of penalties, depending on the memory of the process. For each case, an adaptive estimator is built, and the rates of convergence are computed. Thanks to several sets of simulations, we study the performance of these different penalties for all types of errors (short memory, long memory and anti-persistent errors). Finally, we give an application of our method to the well-known Nile data, which clearly shows that the type of dependence of the error process must be taken into account.

math.ST↗
arXiv2019

Linear regression with stationary errors : the R package slm

This paper introduces the R package slm which stands for Stationary Linear Models. The package contains a set of statistical procedures for linear regression in the general context where the error process is strictly stationary with short memory. We work in the setting of Hannan (1973), who proved the asymptotic normality of the (normalized) least squares estimators (LSE) under very mild conditions on the error process. We propose different ways to estimate the asymptotic covariance matrix of the LSE, and then to correct the type I error rates of the usual tests on the parameters (as well as confidence intervals). The procedures are evaluated through different sets of simulations, and two examples of real datasets are studied.

stat.AP↗
arXiv2019

Rates of convergence in invariance principles for random walks on linear groups via martingale methods

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of Wasserstein distance and in the almost sure invariance principle for R d-valued martingales with stationary increments having moments of order p $\in$]2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups.

math.PR↗
arXiv2019

Criteria for Borel-Cantelli lemmas with applications to Markov chains and dynamical systems

Let (X k) be a strictly stationary sequence of random variables with values in some Polish space E and common marginal $μ$, and (A k) k>0 be a sequence of Borel sets in E. In this paper, we give some conditions on (X k) and (A k) under which the events {X k $\in$ A k } satisfy the Borel-Cantelli (or strong Borel-Cantelli) property. In particular we prove that, if $μ$(lim sup n A n) > 0, the Borel-Cantelli property holds for any absolutely regular sequence. In case where the A k 's are nested, we show, on some examples, that a rate of convergence of the mixing coefficients is needed. Finally we give extensions of these results to weaker notions of dependence, yielding applications to non-irreducible Markov chains and dynamical systems.

math.PR↗