arXiv subjects

Martial Longla

Publications and source records attributed to Martial Longla.

17 recordsLinked to original sources

arXiv2026

New Confidence Regions for Linear Regression Parameters with Stationary-Ergodic Dependent Errors

We develop joint confidence regions for linear regression coefficients when the regressors and errors are jointly stationary and ergodic with unspecified serial dependence. The method applies random smoothing, using an independent auxiliary sample and shrinking bandwidth, to a vector of regression and second-moment statistics. Under stationarity, ergodicity, and finite second moments, the estimator is asymptotically normal and yields Wald confidence regions and simultaneous confidence intervals without direct long-run variance estimation or a parametric dependence model. For implementation, we introduce a scaled estimator with data-driven bandwidth selection and a mild truncation that improves finite-sample stability. Simulations under ARMA, ARFIMA, copula-based Markov errors, and fractional Gaussian noise, with Gaussian and heavy-tailed margins, show near-nominal coverage and competitive region volumes relative to Newey-West HAC and MAC. A winter Beijing PM2.5 application illustrates the procedure. Keywords: Random smoothing, Joint inference, Confidence regions, Dependent errors, Long memory, Regression inference

stat.ME↗

arXiv2025

An Extension of the d-Variate FGM Copula with Application

We introduce an extended d-variate Farlie-Gumbel-Morgenstern (FGM) copula that incorporates additional parameters based on Legendre polynomials to enhance the representation of multivariate dependence structures. Within an i.i.d. framework, we derive closed-form estimators for these parameters and establish their unbiasedness, consistency, and asymptotic normality. A simulation study illustrates the finite-sample performance of the estimators. The model is applied to the Bearing dataset, previously studied by Ota and Kimura (2021) through a d-variate FGM copula and by Longla and Mous-Abou (2025) using an extended bivariate FGM copula. Our analysis shows that the classical d-variate FGM copula does not adequately represent the dependence in this dataset. Based on estimation results and model selection criteria, we propose a reduced version of the extended model as a more appropriate copula specification for the Bearing data.

stat.ME↗

arXiv2024

A Point on Discrete versus Continuous State-Space Markov Chains

This paper examines the impact of discrete marginal distributions on copula-based Markov chains. We present results on mixing and parameter estimation for a copula-based Markov chain model with Bernoulli($p$) marginal distribution and highlight the differences between continuous and discrete state-space Markov chains. We derive estimators for model parameters using the maximum likelihood approach and discuss other estimators of $p$ that are asymptotically equivalent to its maximum likelihood estimator. The asymptotic distributions of the parameter estimators are provided. A simulation study showcases the performance of the different estimators of $p$. Additionally, statistical tests for model parameters are included.

math.ST↗

arXiv2024

New copula families and mixing properties

We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie-Gumbel-Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate $ψ$-mixing Markov chains. Some general results on $ψ$-mixing are given. The Spearman's correlation $ρ_S$ and Kendall's $τ$ are provided for the created copula families. Some general remarks are provided for $ρ_S$ and $τ$. A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.

math.ST↗

arXiv2023

Estimation problems for some perturbations of the independence copula

This work provides a study of parameter estimators based on functions of Markov chains generated by some perturbations of the independence copula. We provide asymptotic distributions of maximum likelihood estimators and confidence intervals for copula parameters of several families of copulas introduced in Longla (2023). Another set of moment-like estimators is proposed along with a multivariate central limit theorem, that provides their asymptotic distributions. We investigate the particular case of Markov chains generated by sine copulas, sine-cosine copulas and the extended Farlie-Gumbel-Morgenstern copula family. Some tests of independence are proposed. A simulation study is provided for the three copula families of interest. This simulation proposes a comparative study of the two introduced estimators and the robust estimator of Longla and Peligrad (2021), showing advantages of the proposed work.

math.ST↗

arXiv2021

On some mixing properties of copula-based Markov chains

This paper brings some insights of $ψ'$-mixing, $ψ^*$-mixing and $ψ$-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for $ψ$-mixing or $ψ'$-mixing, and also show that perturbations of $ψ'$-mixing copula-based Markov chains are $ψ'$-mixing while perturbations of $ψ$-mixing Markov chains are not necessarily $ψ$-mixing Markov chains, even when the perturbed copula is $ψ-mixing$. Some examples of copula families are considered. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains. Moreover, we provide a correction to a statement made in Longla and al. (2021) on $ψ$-mixing.

math.ST↗

arXiv2021

Perturbations of copulas and Mixing properties

This paper explores the impact of perturbations of copulas on the dependence properties of the Markov chains they generate. We consider Markov chains generated by perturbed copulas. Results are provided for the mixing coefficients $β_n$, $ψ_n$ and $ϕ_n$. Several results are provided on mixing for the considered perturbations. New copula functions are provided in connection with perturbations of variables that induce other types of perturbation of copulas not considered in the literature.

math.PR↗

arXiv2021

Dependence and mixing for perturbations of copula-based Markov chains

This paper explores the impact of perturbations of copulas on dependence properties of the Markov chains they generate. We use an observation that is valid for convex combinations of copulas to establish sufficient conditions for the mixing coefficients $ρ_n$, $α_n$ and some other measures of association. New copula families are derived based on perturbations of copulas and their multivariate analogs for $n$-copulas are provided in general. Several families of copulas can be constructed from the provided framework.

math.ST↗

arXiv2018

Remarks on limit theorems for reversible Markov processes

We propose some backward-forward martingale decompositions for functions of reversible Markov chains. These decompositions are used to prove the functional CLT for reversible Markov chains with asymptotically linear variance of partial sums. We also provide a proof of the equivalence between asymptotic linearity of the variance and convergence of the integral of $1/(1-t)$ with respect to the associated spectral measure $ρ$. We also study the asymptotic behavior of linear processes having as innovations mean zero square integrable functions of stationary reversible Markov chains. We apply this study to several cases of reversible stationary Markov chains that arise in regression estimation.

math.PR↗

arXiv2018

On a statistical approach to mate choices in reproduction

We provide a probabilistic approach to modeling the movements of subjects through multiple stages, with "stays" or survival at each stage for a random length of time, and ending at a desired final stage. We use conditional Markov chains with exponential survival times to model the movement of each subject. This is motivated by a study to learn about of the choices that different types of female turkeys make in choosing a male turkey, and in particular, the differences in male choices between groups of females. In this paper, we propose a model for the subjects' movements toward the final stage, and provide maximum likelihood estimation of the model parameters. We also provide results relating to certain questions of interest, such as the distribution of the number of subjects reaching a stage and the probability that a subject reaches the final stage, and develop methods for estimating these quantities and testing statistical hypotheses of interest.

stat.ME↗

arXiv2017

New robust confidence intervals for the mean under dependence

The goal of this paper is to indicate a new method for constructing normal confidence intervals for the mean, when the data is coming from stochastic structures with possibly long memory, especially when the dependence structure is not known or even the existence of the density function. More precisely we introduce a random smoothing suggested by the kernel estimators for the regression function. Applications are presented to linear processes and reversible Markov chains with long memory.

stat.ME↗

arXiv2015

On mixtures of copulas and mixing coefficients

We show that if the density of the absolutely continuous part of a copula is bounded away from zero on a set of Lebesgue measure 1, then that copula generates \textquotedblleft lower $ψ$-mixing\textquotedblright\ stationary Markov chains. This conclusion implies $ϕ$-mixing, $ρ$-mixing, $β$-mixing and \textquotedblleft interlaced $ρ$-mixing\textquotedblright . We also provide some new results on the mixing structure of Markov chains generated by mixtures of copulas.

math.PR↗

arXiv2015

On kernel estimators of density for reversible Markov chains

In this paper we investigate the kernel estimator of the density for a stationary reversible Markov chain. The proofs are based on a new central limit theorem for a triangular array of reversible Markov chains obtained under conditions imposed to covariances, which has interest in itself.

math.PR↗

arXiv2013

On Dependence Structure of Copula-based Markov chains

We consider dependence coefficients for stationary Markov chains. We emphasize on some equivalencies for reversible Markov chains. We improve some known results and provide a necessary condition for Markov chains based on Archimedean copulas to be exponential $ρ$-mixing. We analyze the example of the Mardia and Frechet copula families using small sets.

math.ST↗

arXiv2013

On Functional CLT for Reversible Markov Chains with nonlinear growth of the Variance

In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

math.PR↗

arXiv2013

Remarks on the speed of convergence of mixing coefficients and applications

In this paper, we study dependence coefficients for copula-based Markov chains. We provide new tools to check the convergence rates of mixing coefficients of copula-based Markov chains. We study Markov chains generated by the Metropolis-hastings algorithm and give conditions on the proposal that ensure exponential $ρ$-mixing, $β$-mixing and $ϕ$-mixing. A general necessary condition on symmetric copulas to generate exponential $ρ$-mixing or $ϕ$-mixing is given. At the end of the paper, we comment and improve some of our previous results on mixtures of copulas.

math.PR↗

arXiv2012

Some Aspects of Modeling Dependence in Copula-based Markov chains

Dependence coefficients have been widely studied for Markov processes defined by a set of transition probabilities and an initial distribution. This work clarifies some aspects of the theory of dependence structure of Markov chains generated by copulas that are useful in time series econometrics and other applied fields. The main aim of this paper is to clarify the relationship between the notions of geometric ergodicity and geometric ρ-mixing; namely, to point out that for a large number of well known copulas, such as Clayton, Gumbel or Student, these notions are equivalent. Some of the results published in the last years appear to be redundant if one takes into account this fact. We apply this equivalence to show that any mixture of Clayton, Gumbel or Student copulas generate both geometrically ergodic and geometric ρ-mixing stationary Markov chains, answering in this way an open question in the literature. We shall also point out that a sufficient condition for ρ-mixing, used in the literature, actually implies Doeblin recurrence.

math.PR↗