arXiv subjects

Éric Marchand

Publications and source records attributed to Éric Marchand.

17 recordsLinked to original sources

arXiv2026

On the Bayesian analysis of a non-identifiable Binomial model

We provide analysis for the posterior distribution and expectation of $(p_1, p_2)$ where $Y|p_1,p_2 \sim \hbox{Binomial}(n, p_1 p_2)$ and $ (p_1, p_2)$ is uniformly distributed on the unit square $[0,1]^2$. We exhibit interesting expressions in terms of a truncated Beta distribution, a finite mixture of Beta distributions and harmonic numbers, and derive a simple large sample size $n$ approximation for the posterior expectations $\mathbb{E}(p_i|y)$ as well as for the normalization constant in the posterior joint density.

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arXiv2025

Tree Pólya Splitting distributions for multivariate count data

In this article, we develop a new class of multivariate distributions adapted for count data, called Tree Pólya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. Known distributions, including the Dirichlet-multinomial, the generalized Dirichlet-multinomial and the Dirichlet-tree multinomial, are particular cases within this class. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependence structures (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree Pólya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependence structures (covariance and correlations). A dataset of abundance of Trichoptera is used, on one hand, as a benchmark to illustrate the theoretical properties developed in this article, and on the other hand, to demonstrate the interest of these types of models, notably by comparing them to other approaches for fitting multivariate data, such as the Poisson-lognormal model in ecology or singular multivariate distributions used in microbiome.

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arXiv2024

Bayesian prediction regions and density estimation with type-2 censored data

For exponentially distributed lifetimes, we consider the prediction of future order statistics based on having observed the first $m$ order statistics. We focus on the previously less explored aspects of predicting: (i) an arbitrary pair of future order statistics such as the next and last ones, as well as (ii) the next $N$ future order statistics. We provide explicit and exact Bayesian credible regions associated with Gamma priors, and constructed by identifying a region with a given credibility $1-λ$ under the Bayesian predictive density. For (ii), the HPD region is obtained, while a two-step algorithm is given for (i). The predictive distributions are represented as mixtures of bivariate Pareto distributions, as well as multivariate Pareto distributions. For the non-informative prior density choice, we demonstrate that a resulting Bayesian credible region has matching frequentist coverage probability, and that the resulting predictive density possesses the optimality properties of best invariance and minimaxity.

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arXiv2023

Bayesian and minimax estimators of loss

We study the problem of loss estimation that involves for an observable $X \sim f_θ$ the choice of a first-stage estimator $\hatγ$ of $γ(θ)$, incurred loss $L=L(θ, \hatγ)$, and the choice of a second-stage estimator $\hat{L}$ of $L$. We consider both: (i) a sequential version where the first-stage estimate and loss are fixed and optimization is performed at the second-stage level, and (ii) a simultaneous version with a Rukhin-type loss function designed for the evaluation of $(\hatγ, \hat{L})$ as an estimator of $(γ, L)$. We explore various Bayesian solutions and provide minimax estimators for both situations (i) and (ii). The analysis is carried out for several probability models, including multivariate normal models $N_d(θ, σ^2 I_d)$ with both known and unknown $σ^2$, Gamma, univariate and multivariate Poisson, and negative binomial models, and relates to different choices of the first-stage and second-stage losses. The minimax findings are achieved by identifying least favourable of sequence of priors and depend critically on particular Bayesian solution properties, namely situations where the second-stage estimator $\hat{L}(x)$ is constant as a function of $x$.

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arXiv2023

Asymptotic tail properties of Poisson mixture distributions

Count data are omnipresent in many applied fields, often with overdispersion. With mixtures of Poisson distributions representing an elegant and appealing modelling strategy, we focus here on how the tail behaviour of the mixing distribution is related to the tail of the resulting Poisson mixture. We define five sets of mixing distributions and we identify for each case whenever the Poisson mixture is in, close to or far from a domain of attraction of maxima. We also characterize how the Poisson mixture behaves similarly to a standard Poisson distribution when the mixing distribution has a finite support. Finally, we study, both analytically and numerically, how goodness-of-fit can be assessed with the inspection of tail behaviour.

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arXiv2021

On shrinkage estimation of a spherically symmetric distribution for balanced loss functions

We consider the problem of estimating the mean vector $θ$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $ωρ(\|\de-\de_{0}\|^{2}) +(1-ω)ρ(\|\de - θ\|^{2})$ and {\bf (ii)} $\ell\left(ω\|\de - \de_{0}\|^{2} +(1-ω)\|\de - θ\|^{2}\right)$, where $δ_0$ is a target estimator, and where $ρ$ and $\ell$ are increasing and concave functions. For $d\geq 4$ and the target estimator $δ_0(X)=X$, we provide Baranchik-type estimators that dominate $δ_0(X)=X$ and are minimax. The findings represent extensions of those of Marchand \& Strawderman (\cite{ms2020}) in two directions: {\bf (a)} from scale mixture of normals to the spherical class of distributions with Lebesgue densities and {\bf (b)} from completely monotone to concave $ρ'$ and $\ell'$.

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arXiv2020

Bayesian estimation and prediction for certain mixtures

For two vast families of mixture distributions and a given prior, we provide unified representations of posterior and predictive distributions. Model applications presented include bivariate mixtures of Gamma distributions labelled as Kibble-type, non-central Chi-square and F distributions, the distribution of $R^2$ in multiple regression, variance mixture of normal distributions, and mixtures of location-scale exponential distributions including the multivariate Lomax distribution. An emphasis is also placed on analytical representations and the relationships with a host of existing distributions and several hypergeomtric functions of one or two variables.

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arXiv2019

On shrinkage estimation for balanced loss functions

The estimation of a multivariate mean $θ$ is considered under natural modifications of balanced loss function of the form: (i) $ω\, ρ(\|δ-δ_0\|^2) + (1-ω) \, ρ(\|δ-θ\|^2) $, and (ii) $\ell \left( ω\, \|δ-δ_0\|^2 + (1-ω) \, \|δ-θ\|^2 \right)\,$, where $δ_0$ is a target estimator of $γ(θ)$. After briefly reviewing known results for original balanced loss with identity $ρ$ or $\ell$, we provide, for increasing and concave $ρ$ and $\ell$ which also satisfy a completely monotone property, Baranchik-type estimators of $θ$ which dominate the benchmark $δ_0(X)=X$ for $X$ either distributed as multivariate normal or as a scale mixture of normals. Implications are given with respect to model robustness and simultaneous dominance with respect to either $ρ$ or $\ell

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arXiv2018

On efficient prediction and predictive density estimation for spherically symmetric models

Let $X,U,Y$ be spherically symmetric distributed having density $$η^{d +k/2} \, f\left(η(\|x-θ|^2+ \|u\|^2 + \|y-cθ\|^2 ) \right)\,,$$ with unknown parameters $θ\in \mathbb{R}^d$ and $η>0$, and with known density $f$ and constant $c >0$. Based on observing $X=x,U=u$, we consider the problem of obtaining a predictive density $\hat{q}(y;x,u)$ for $Y$ as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant density $\hat{q}_{mre}$, which is Generalized Bayes with respect to the prior $π(θ, η) = η^{-1}$. For $d \geq 3$, we obtain improvements on $\hat{q}_{mre}$, and further show that the dominance holds simultaneously for all $f$ subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior $π_h(θ, η) =η^{-1} \|θ\|^{2-d}$ dominates $\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.

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arXiv2018

Inference for a constrained parameter in presence of an uncertain constraint

We describe a hierarchical Bayesian approach for inference about a parameter $θ$ lower-bounded by $α$ with uncertain $α$, derive some basic identities for posterior analysis about $(θ,α)$, and provide illustrations for normal and Poisson models. For the normal case with unknown mean $θ$ and known variance $σ^2$, we obtain Bayes estimators of $θ$ that take values on $\mathbb{R}$, but that are equally adapted to a lower-bound constraint in being minimax under squared error loss for the constrained problem.

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arXiv2018

On Predictive Density Estimation under $α$-divergence Loss

Based on $X \sim N_d(θ, σ^2_X I_d)$, we study the efficiency of predictive densities under $α-$divergence loss $L_α$ for estimating the density of $Y \sim N_d(θ, σ^2_Y I_d)$. We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension $d$, the variances $σ^2_X$ and $σ^2_Y$, the choice of loss $L_α$; $α\in (-1,1)$. The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with $θ\in Θ\subset \mathbb{R}^d$. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.

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arXiv2017

On predictive density estimation with additional information

Based on independently distributed $X_1 \sim N_p(θ_1, σ^2_1 I_p)$ and $X_2 \sim N_p(θ_2, σ^2_2 I_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(θ_1, σ^2_Y I_p)$, with the additional information $θ_1 - θ_2 \in A$ and known $σ^2_1, σ^2_2, σ^2_Y$. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for $α-$divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss in the univariate case ($p=1$). An ensemble of techniques are exploited, including variance expansion (for KL loss), point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for $\hat{q}_{π_{U,A}}$ associated with a uniform prior for $θ=(θ_1, θ_2)$ truncated to $\{θ\in \mathbb{R}^{2p}: θ_1 - θ_2 \in A \}$, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.

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arXiv2014

On Predictive Density Estimation for Location Families under Integrated $L_2$ and $L_1$ Losses

Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequentist risk of such predictive densities with integrated $L_2$ and $L_1$ losses. Our findings relate to a $p-$variate spherically symmetric observable $X \sim p_X(\|x-μ\|^2)$ and the objective of estimating the density of $Y \sim q_Y(\|y-μ\|^2)$ based on $X$. For $L_2$ loss, we describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three of more dimensions (i.e., $p \geq 3$), we show that the MRE estimator is inadmissible under $L_2$ loss and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of $\|\hatμ-μ\|^2$. We also study the phenomenon of improvement on the plug-in density estimator of the form $q_Y(\|y-aX\|^2)\,, 0 1$, showing in some cases, inevitably for large enough $p$, that all choices $c>1$ are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for $p \geq 3$. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with $p \geq 4$ under $L_1$ loss.

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arXiv2012

A unified minimax result for restricted parameter spaces

We provide a development that unifies, simplifies and extends considerably a number of minimax results in the restricted parameter space literature. Various applications follow, such as that of estimating location or scale parameters under a lower (or upper) bound restriction, location parameter vectors restricted to a polyhedral cone, scale parameters subject to restricted ratios or products, linear combinations of restricted location parameters, location parameters bounded to an interval with unknown scale, quantiles for location-scale families with parametric restrictions and restricted covariance matrices.

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arXiv2012

Estimation of a multivariate normal mean with a bounded signal to noise ratio

For normal canonical models with $X \sim N_p(θ, σ^{2} I_{p}), \;\; S^{2} \sim σ^{2}χ^{2}_{k}, \;{independent}$, we consider the problem of estimating $θ$ under scale invariant squared error loss $\frac{\|d-θ\|^{2}}{σ^{2}}$, when it is known that the signal-to-noise ratio $\frac{\|θ\|}σ$ is bounded above by $m$. Risk analysis is achieved by making use of a conditional risk decomposition and we obtain in particular sufficient conditions for an estimator to dominate either the unbiased estimator $δ_{UB}(X)=X$, or the maximum likelihood estimator $δ_{\hbox{mle}}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator $δ_{BU}$ associated with a prior on $(θ,σ)$ such that $θ|σ$ is uniformly distributed on the (boundary) sphere of radius $m$ and a non-informative $\frac{1}σ$ prior measure is placed marginally on $σ$. With a series of technical results related to $δ_{BU}$; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m \leq \sqrt{p}$ and $p \geq 2$, $δ_{BU}$ dominates both $δ_{UB}$ and $δ_{\hbox{mle}}$. The finding can be viewed as both a multivariate extension of $p=1$ result due to Kubokawa (2005) and a unknown variance extension of a similar dominance finding due to Marchand and Perron (2001). Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for $m \leq \sqrt{\frac{p}{2}}$, a wide class of Bayes estimators, which include priors where $θ|σ$ is uniformly distributed on the ball of radius $m$, are shown to dominate $δ_{UB}$.

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arXiv2008

On the frequentist coverage of Bayesian credible intervals for lower bounded means

For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-α)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\frac{3α}{2}$ and $1-\frac{3α}{2}+\frac{α^2}{1+α}$; with the lower bound $1-\frac{3α}{2}$ improving (for $α\leq 1/3$) on the previously established ([9]; [8]) lower bound $\frac{1-α}{1+α}$. Several illustrative examples are given.

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arXiv2006

On the behavior of Bayesian credible intervals for some restricted parameter space problems

For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-α)%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the restricted parameter space. Namely, they establish the attractive lower bound of $\frac{1-α}{1+α}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-α}{1+α}$ is applicable for a substantially more general setting with underlying distributional symmetry, and obtain various other properties. The derivations are unified and are driven by the choice of a right Haar invariant prior. Investigations of non-symmetric models are carried out and similar results are obtained. Namely, (i) we show that the lower bound $\frac{1-α}{1+α}$ still applies for certain types of asymmetry (or skewness), and (ii) we extend results obtained by Zhang and Woodroofe (2002) for estimating the scale parameter of a Fisher distribution; which arises in estimating the ratio of variance components in a one-way balanced random effects ANOVA. Finally, various examples illustrating the wide scope of applications are expanded upon. Examples include estimating parameters in location models and location-scale models, estimating scale parameters in scale models, estimating linear combinations of location parameters such as differences, estimating ratios of scale parameters, and problems with non-independent observations.

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