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In this article, we develop a new class of multivariate distributions adapted for count data, called Tree P{\'o}lya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependencies (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree P{\'o}lya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependency structures (covariance and correlations). The abundance of 17 species of Trichoptera recorded at 49 sites is used, on one hand, to illustrate the theoretical properties developed in this article on a concrete case, and on the other hand, to demonstrate the interest of this type of models, notably by comparing them to classical approaches in ecology or microbiome.

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 $1-\lambda$ 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., Comment: 15 pages, 4 figures

We study the problem of loss estimation that involves for an observable $X \sim f_{\theta}$ the choice of a first-stage estimator $\hat{\gamma}$ of $\gamma(\theta)$, incurred loss $L=L(\theta, \hat{\gamma})$, 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 $(\hat{\gamma}, \hat{L})$ as an estimator of $(\gamma, L)$. 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 $N_d(\theta, \sigma^2 I_d)$ with both known and unknown $\sigma^2$, 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 $x$., Comment: 27 pages

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., Comment: Stat, In press

This paper addresses the problem of an efficient predictive density estimation for the density $q(\|y-\theta\|^2)$ of $Y$ based on $X \sim p(\|x-\theta\|^2)$ for $y, x, \theta \in \mathbb{R}^d$. The chosen criteria are integrated $L_1$ loss given by $L(\theta, \hat{q}) \, =\, \int_{\mathbb{R}^d} \big|\hat{q}(y)- q(\|y-\theta\|^2) \big| \, dy$, and the associated frequentist risk, for $\theta \in \Theta$. For absolutely continuous and strictly decreasing $q$, we establish the inevitability of scale expansion improvements $\hat{q}_c(y;X)\,=\, \frac{1}{c^d} q\big(\|y-X\|^2/c^2 \big) $ over the plug-in density $\hat{q}_1$, for a subset of values $c \in (1,c_0)$. The finding is universal with respect to $p,q$, and $d \geq 2$, and extended to loss functions $\gamma \big(L(\theta, \hat{q} ) \big)$ with strictly increasing $\gamma$. The finding is also extended to include scale expansion improvements of more general plug-in densities $q(\|y-\hat{\theta}(X)\|^2 \big)$, when the parameter space $\Theta$ is a compact subset of $\mathbb{R}^d$. Numerical analyses illustrative of the dominance findings are presented and commented upon. As a complement, we demonstrate that the unimodal assumption on $q$ is necessary with a detailed analysis of cases where the distribution of $Y|\theta$ is uniformly distributed on a ball centered about $\theta$. In such cases, we provide a univariate ($d=1$) example where the best equivariant estimator is a plug-in estimator, and we obtain cases (for $d=1,3$) where the plug-in density $\hat{q}_1$ is optimal among all $\hat{q}_c$., Comment: 20 pages, 3 figures

We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter $\theta \in \mathbb{R}^d$, and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for $\theta$. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on $\theta$, and illustrate and comment the findings based on numerical evaluations., Comment: 25 pages,2 figures

We consider the problem of estimating the mean vector $\theta$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $\omega \rho(\|\de-\de_{0}\|^{2}) +(1-\omega)\rho(\|\de - \theta\|^{2})$ and {\bf (ii)} $\ell\left(\omega \|\de - \de_{0}\|^{2} +(1-\omega)\|\de - \theta\|^{2}\right)$, where $\delta_0$ is a target estimator, and where $\rho$ and $\ell$ are increasing and concave functions. For $d\geq 4$ and the target estimator $\delta_0(X)=X$, we provide Baranchik-type estimators that dominate $\delta_0(X)=X$ 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 $\rho'$ and $\ell'$.

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., Comment: 23 pages, 1 figure

The estimation of a multivariate mean $\theta$ is considered under natural modifications of balanced loss function of the form: (i) $\omega \, \rho(\|\delta-\delta_0\|^2) + (1-\omega) \, \rho(\|\delta-\theta\|^2) $, and (ii) $\ell \left( \omega \, \|\delta-\delta_0\|^2 + (1-\omega) \, \|\delta-\theta\|^2 \right)\,$, where $\delta_0$ is a target estimator of $\gamma(\theta)$. After briefly reviewing known results for original balanced loss with identity $\rho$ or $\ell$, we provide, for increasing and concave $\rho$ and $\ell$ which also satisfy a completely monotone property, Baranchik-type estimators of $\theta$ which dominate the benchmark $\delta_0(X)=X$ 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 $\rho$ or $\ell, Comment: 15 pages

Let $X,U,Y$ be spherically symmetric distributed having density $$\eta^{d +k/2} \, f\left(\eta(\|x-\theta|^2+ \|u\|^2 + \|y-c\theta\|^2 ) \right)\,,$$ with unknown parameters $\theta \in \mathbb{R}^d$ and $\eta>0$, 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 $\pi(\theta, \eta) = \eta^{-1}$. 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 $\pi_h(\theta, \eta) =\eta^{-1} \|\theta\|^{2-d}$ dominates $\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.

We describe a hierarchical Bayesian approach for inference about a parameter $\theta$ lower-bounded by $\alpha$ with uncertain $\alpha$, derive some basic identities for posterior analysis about $(\theta,\alpha)$, and provide illustrations for normal and Poisson models. For the normal case with unknown mean $\theta$ and known variance $\sigma^2$, we obtain Bayes estimators of $\theta$ 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., Comment: 10 pages, 1 figure

Based on $X \sim N_d(\theta, \sigma^2_X I_d)$, we study the efficiency of predictive densities under $\alpha-$divergence loss $L_{\alpha}$ for estimating the density of $Y \sim N_d(\theta, \sigma^2_Y I_d)$. 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 $\sigma^2_X$ and $\sigma^2_Y$, the choice of loss $L_{\alpha}$; $\alpha \in (-1,1)$. The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with $\theta \in \Theta \subset \mathbb{R}^d$. 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., Comment: 19 pages, 5 Figures

Based on independently distributed $X_1 \sim N_p(\theta_1, \sigma^2_1 I_p)$ and $X_2 \sim N_p(\theta_2, \sigma^2_2 I_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(\theta_1, \sigma^2_Y I_p)$, with the additional information $\theta_1 - \theta_2 \in A$ and known $\sigma^2_1, \sigma^2_2, \sigma^2_Y$. 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 $\alpha-$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 $\hat{q}_{\pi_{U,A}}$ associated with a uniform prior for $\theta=(\theta_1, \theta_2)$ truncated to $\{\theta \in \mathbb{R}^{2p}: \theta_1 - \theta_2 \in A \}$, 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., Comment: 30 pages, 4 Figures

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 $X \sim p_X(\|x-\mu\|^2)$ and the objective of estimating the density of $Y \sim q_Y(\|y-\mu\|^2)$ 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 $\|\hat{\mu}-\mu\|^2$. We also study the phenomenon of improvement on the plug-in density estimator of the form $q_Y(\|y-aX\|^2)\,, 01$, 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.

We consider the problem of estimating a continuous distribution function $F$, as well as meaningful functions $\tau(F)$ under a large class of loss functions. We obtain best invariant estimators and establish their minimaxity for H\"{o}lder continuous $\tau$'s and strict bowl-shaped losses with a bounded derivative. We also introduce and motivate the use of integrated balanced loss functions which combine the criteria of an integrated distance between a decision $d$ and $F$, with the proximity of $d$ with a target estimator $d_0$. Moreover, we show how the risk analysis of procedures under such an integrated balanced loss relates to a dual risk analysis under an "unbalanced" loss, and we derive best invariant estimators, minimax estimators, risk comparisons, dominance and inadmissibility results. Finally, we expand on various illustrations and applications relative to maxima-nomination sampling, median-nomination sampling, and a case study related to bilirubin levels in the blood of babies suffering from jaundice., Comment: 22pages, 3 figures, 1 table

For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-\alpha$ and frequentist coverage probability bounded below by $\frac{1-\alpha}{1+\alpha}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a such a class of Bayesian credible sets fills a gap in the literature and includes an "equal-tails" modification of the HPD procedure. Several examples are presented demonstrating wide applicability., Comment: This is an expanded version of an earlier post

For normal canonical models, and more generally a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant $L^p$ loss with $p>0$. We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa's integral expression for risk difference technique. Several other analytical properties are obtained, including a robustness property of the generalized Bayes estimators above when the loss is either scale invariant $L^p$ or asymmetrized versions. Applications include inference in two-sample normal model with order constraints on the means., Comment: 21 pages

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., Comment: Published in at http://dx.doi.org/10.3150/10-BEJ336 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

For normal canonical models with $X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;{independent}$, we consider the problem of estimating $\theta$ under scale invariant squared error loss $\frac{\|d-\theta \|^{2}}{\sigma^{2}}$, when it is known that the signal-to-noise ratio $\frac{\|\theta\|}{\sigma}$ 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 $\delta_{UB}(X)=X$, or the maximum likelihood estimator $\delta_{\hbox{mle}}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator $\delta_{BU}$ associated with a prior on $(\theta,\sigma)$ such that $\theta|\sigma$ is uniformly distributed on the (boundary) sphere of radius $m$ and a non-informative $\frac{1}{\sigma}$ prior measure is placed marginally on $\sigma$. With a series of technical results related to $\delta_{BU}$; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m \leq \sqrt{p}$ and $p \geq 2$, $\delta_{BU}$ dominates both $\delta_{UB}$ and $\delta_{\hbox{mle}}$. 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 $\theta|\sigma$ is uniformly distributed on the ball of radius $m$, are shown to dominate $\delta_{UB}$.

For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-\alpha)%$ 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 $1-\frac{3\alpha}{2}$ and $1-\frac{3\alpha}{2}+\frac{\alpha^2}{1+\alpha}$; with the lower bound $1-\frac{3\alpha}{2}$ improving (for $\alpha \leq 1/3$) on the previously established ([9]; [8]) lower bound $\frac{1-\alpha}{1+\alpha}$. Several illustrative examples are given., Comment: Published in at http://dx.doi.org/10.1214/08-EJS292 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-\alpha)%$ 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 $\frac{1-\alpha}{1+\alpha}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-\alpha}{1+\alpha}$ 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 $\frac{1-\alpha}{1+\alpha}$ 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., Comment: Published at http://dx.doi.org/10.1214/074921706000000635 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)