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Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $ \Sigma.$ Randomize $\Sigma$ such that $ \Sigma^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)/2$ and mean $p\sigma.$ We compute the density $f_{p,\sigma}$ of $X$ as well as the Fisher information $I_p(\sigma)$ of the model $(f_{p,\sigma} )$ when $\sigma $ is the parameter. For using the Cram\'er-Rao inequality, we also compute the inverse of $I_p(\sigma)$. The important point of this note is the fact that this inverse is a linear combination of two simple operators on the space of symmetric matrices, namely $\P(\sigma)(s)=\sigma s \sigma$ and $(\sigma\otimes \sigma)(s)=\sigma \, \mathrm{trace}(\sigma s)$. The Fisher information itself is a linear combination $\P(\sigma^{-1})$ and $\sigma^{-1}\otimes \sigma^{-1}.$ Finally, by randomizing $\sigma $ itself, we make explicit the minoration of the second moments of an estimator of $\sigma$ by the Van Trees inequality: here again, linear combinations of $\P(u)$ and $u\otimes u$ appear in the results., Comment: 11 pages

If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B} $$ are independent, we prove that this happens if and only if the $A$ and $B$ have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack (2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor (2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg - de Vries equation, where the above result was conjectured. We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open., Comment: 17 pages

Consider a measure $\mu$ on $\R^n$ generating a natural exponential family $F(\mu)$ with variance function $V_{F(\mu)}(m)$ and Laplace transform $$ \exp(\ell_{\mu}(s))=\int_{\R^n} \exp(-\)\mu(dx).$$ A dual measure $\mu^*$ satisfies $-\ell'_{\mu^*}(-\ell'_{\mu}(s))=s.$ Such a dual measure does not always exist. One important property is $\ell"_{\mu^*}(m)=(V_{F(\mu)}(m))^{-1},$ leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families $T\hskip-2pt F$ obtained by considering all translations of a given exponential family $F$)., Comment: 25 pages

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $\alpha$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq \alpha \leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $\mu_\alpha$ such that $P_0$ is the Gaussian distribution $N(\mu_{\alpha},1).$ In other terms, if for all $t$, $t$ is a quantile of $P_t$ associated to some threshold $\alpha\in (0,1)$, then the exponential family must be Gaussian. The case $\alpha=1/2$, \textit{i.e.} $t$ is always a median of $P_t,$ has been considered in Letac \textit{et al.} (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq \alpha \leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_{\alpha}>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma distribution with parameters $p$ and $t.$, Comment: 7 pages

For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix., Comment: 13 pages

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Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^k\mu(du)$ for some number $c_k$ and some positive measure $\mu,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Moebius functions preserve the cross ratio, \textsc{Keywords:} Binomial, Poisson and negative binomial distributions. Gaussian and Gamma distributions. Moebius transforms. Cross ratio., Comment: 10 pages

In Sabot and Tarr\`es (2015), the authors have explicitly computed the integral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\ldots,2x_n)$. The domain of integration is the part of $\mathbb{R}^n$ for which $M_x$ is positive definite. We calculate more generally for $ b_1\geq 0,\ldots b_n\geq 0$ the integral $$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that it leads to a natural family of distributions in $\mathbb{R}^n$, called the $GSTZ_n$ probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. We also show that if the graph with the set of vertices $V=\{1,\ldots,n\}$ and the set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q.$, Comment: 17 pages

Let $P$ a probability on the real line generating a natural exponential family $(P_t)_{t\in \R}$. We show that $t$ is a median of $P_t$ for all $t$ only if $P$ is the standard Gaussian law $N(0.1).$ The proof is based on the Choquet Deny equation., Comment: 5 pages

Consider $G=SL_2(\mathbb{Z})/\{\pm I\}$ acting on the complex upper half plane $H$ by $h_M(z)=\frac{az+b}{cz+d},$ for $M \in G$. Let $D=\{z \in H: |z|\geq 1, |\Re(z)|\leq 1/2\}$. We consider the set $\mathcal{E} \subset G$ with the $9$ elements $M$, different from the identity, such that $(MM^T)\leq 3$. We equip the tiling of $H$ defined by $\mathbb{D}=\{h_M(D), M \in G\}$ with a graph structure where the neighbours are defined by $h_M(D) \cap h_{M'}(D) \neq \emptyset$, equivalently $M^{-1}M' \in \mathcal{E}$. The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point $X$ of the real line with the same distribution of $S_2 W^{S_1}$, where $S_1,S_2,W$ are independent with $\Pr (S_i=\pm 1)=1/2$ and where $W$ is valued in $(0,1)$ with distribution $\Pr(W

Gaussian graphical models are relevant tools to learn conditional independence structure between variables. In this class of models, Bayesian structure learning is often done by search algorithms over the graph space. The conjugate prior for the precision matrix satisfying graphical constraints is the well-known G-Wishart. With this prior, the transition probabilities in the search algorithms necessitate evaluating the ratios of the prior normalizing constants of G-Wishart. In moderate to high-dimensions, this ratio is often approximated using sampling-based methods as computationally expensive updates in the search algorithm. Calculating this ratio so far has been a major computational bottleneck. We overcome this issue by representing a search algorithm in which the ratio of normalizing constant is carried out by an explicit closed-form approximation. Using this approximation within our search algorithm yields significant improvement in the scalability of structure learning without sacrificing structure learning accuracy. We study the conditions under which the approximation is valid. We also evaluate the efficacy of our method with simulation studies. We show that the new search algorithm with our approximation outperforms state-of-the-art methods in both computational efficiency and accuracy. The implementation of our work is available in the R package BDgraph., Comment: 59 pages

If $\alpha$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the Dirichlet random probability $P_t\sim\mathcal{D}(t\alpha) ;$ it is such that for any measurable partition $(A_0,\ldots,A_k)$ of $\mathbb{R}^d$ then $(P_t(A_0),\ldots,P_t(A_k))$ is Dirichlet distributed with parameters $(t\alpha(A_0)\ldots,t\alpha(A_k)).$ If $\int_{\mathbb{R}^d}\log(1+\|x\|)\alpha(dx)<\infty$ the random variable $\int_{\mathbb{R}^d}xP_t(dx)$ of $\mathbb{R}^d$ does exist and we denote by $\mu(t\alpha)$ its distribution. The Dirichlet curve associated to the probability $\alpha$ is the map $t\mapsto \mu(t\alpha).$ It has simple properties like $\lim_{t\searrow 0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow \infty}\mu(t\alpha)=\delta_m$ when $m=\int_{\mathbb{R}^d} x\alpha(dx)$ exists. The present paper shows first that if $m$ exists and if $\psi$ is a convex function on $\mathbb{R}^d$ then $t\mapsto \int_{\mathbb{R}^d}\psi(x)\mu(t\alpha)(dx)$ is a decreasing function, which means that $t\mapsto \mu(t\alpha)$ is decreasing according to the Strassen convex order of probabilities. The second aim of the paper is to prove a group of results around the following question: if $\mu(t\alpha)=\mu(s\alpha)$ for some $0\leq s

This article provides tools for the study of the Dirichlet random walk in $\mathbb{R}^d$. By this we mean the random variable $W=X_1\Theta_1+\cdots+X_n\Theta_n$ where $X=(X_1,\ldots,X_n) \sim \mathcal{D}(q_1,\ldots,q_n)$ is Dirichlet distributed and where $\Theta_1,\ldots \Theta_n$ are iid, uniformly distributed on the unit sphere of $\mathbb{R}^d$ and independent of $X.$ In particular we compute explicitely in a number of cases the distribution of $W.$ Some of our results appear already in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010, 2011). In these cases, our proofs are much simpler from the original ones, since we use a kind of Stieltjes transform of $W$ instead of the Laplace transform: as a consequence the hypergeometric functions replace the Bessel functions. A crucial ingredient is a particular case of the classical and non trivial identity, true for $0\leq u\leq 1/2$:$$_2F_1(2a,2b;a+b+\frac{1}{2};u)= \_2F_1(a,b;a+b+\frac{1}{2};4u-4u^2).$$ We extend these results to a study of the limits of the Dirichlet random walks when the number of added terms goes to infinity, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and of Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit ball of $\mathbb{R}^d.$ {4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms, random flight, distributions in a ball, hyperuniformity, infinite divisibility in the sense of Dirichlet. {4mm}\noindent \textsc{AMS classification}: 60D99, 60F99.

Let $X$, $B$ and $Y$ be three Dirichlet, Bernoulli and beta independent random variables such that $X\sim \mathcal{D}(a_0,...,a_d),$ such that $\Pr(B=(0,...,0,1,0,...,0))=a_i/a$ with $a=\sum_{i=0}^da_i$ and such that $Y\sim \beta(1,a).$ We prove that $X\sim X(1-Y)+BY.$ This gives the stationary distribution of a simple Markov chain on a tetrahedron. We also extend this result to the case when $B$ follows a quasi Bernoulli distribution $\mathcal{B}_k(a_0,...,a_d)$ on the tetrahedron and when $Y\sim \beta(k,a)$. We extend it even more generally to the case where $X$ is a Dirichlet process and $B$ is a quasi Bernoulli random probability. Finally the case where the integer $k$ is replaced by a positive number $c$ is considered when $a_0=...=a_d=1.$ \textsc{Keywords} \textit{Perpetuities, Dirichlet process, Ewens distribution, quasi Bernoulli laws, probabilities on a tetrahedron, $T_c$ transform, stationary distribution.} AMS classification 60J05, 60E99., Comment: 18 pages

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone $\bar{\mathcal{P}_d}$ of semi positive definite matrices of order $d$ and with a real valued shape parameter, exists. We reduce this problem to the problem of existence of the measures $m(n,k,d)$ defined on $\bar{\mathcal{P}_d}$ and with Laplace transform $(\det s)^{-n/2}\exp \tr(s^{-1}w)$ where $n$ is an integer and where $w=\mathrm{diag}(0,...,0,1,...,1)$ has order $d$ and rank $k.$ We compute $m(d-1,d,d)$ and we show that neither $m(d-2,d,d)$ nor $m(d-2,d-1,d)$ exist. This proves a conjecture of E. Mayerhofer.

A standard tool for model selection in a Bayesian framework is the Bayes factor which compares the marginal likelihood of the data under two given different models. In this paper, we consider the class of hierarchical loglinear models for discrete data given under the form of a contingency table with multinomial sampling. We assume that the Diaconis-Ylvisaker conjugate prior is the prior distribution on the loglinear parameters and the uniform is the prior distribution on the space of models. Under these conditions, the Bayes factor between two models is a function of their prior and posterior normalizing constants. These constants are functions of the hyperparameters $(m,\alpha)$ which can be interpreted respectively as marginal counts and the total count of a fictive contingency table. We study the behaviour of the Bayes factor when $\alpha$ tends to zero. In this study two mathematical objects play a most important role. They are, first, the interior $C$ of the convex hull $\bar{C}$ of the support of the multinomial distribution for a given hierarchical loglinear model together with its faces and second, the characteristic function $\mathbb{J}_C$ of this convex set $C$. We show that, when $\alpha$ tends to 0, if the data lies on a face $F_i$ of $\bar{C_i},i=1,2$ of dimension $k_i$, the Bayes factor behaves like $\alpha^{k_1-k_2}$. This implies in particular that when the data is in $C_1$ and in $C_2$, i.e. when $k_i$ equals the dimension of model $J_i$, the sparser model is favored, thus confirming the idea of Bayesian regularization., Comment: 37 pages

We construct a family of matrix ensembles that fits Anshelevich's regression postulates for "Meixner laws on matrices". We show that the Laplace transform of a general n by n Meixner matrix ensemble satisfies a system of partial differential equations which is explicitly solvable for n=2. We rely on these solutions to identify the six types of 2 by 2 Meixner matrix ensembles.

Consider an exponential dispersion model (EDM) generated by a probability $ \mu $ on $[0,\infty )$ which is infinitely divisible with an unbounded L\'{e}vy measure $\nu $. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $\theta _{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(\theta _{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(\theta_0,t)$. Then if $\nu ((x,\infty ))\sim -\ell \log x$ around zero we prove that the limiting law $F_0$ of $ Y_{t}^{-t}$ as $t\rightarrow 0$ is of a Pareto type (not depending on $ \theta_0$) with the form $F_0(u)=0$ for $u<1$ and $1-u^{-\ell }$ for $ u\geq 1$. Such a result enables an approximation of the distribution of $ Y_{t}$ for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided., Comment: 8 pages

Hazard rates play an important role in various areas, e.g., reliability theory, survival analysis, biostatistics, queueing theory and actuarial studies. Mixtures of distributions are also of a great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements\ of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEF's having either quadratic, or cubic variance functions (VF) and others as well. A particular attention is devoted to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF and to the Kummer distributions of type 2 NEF. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEF's. Various lemmas and propositions are needed then to verify this condition for these NEF's., Comment: 21 pages

If the space $\mathcal{Q}$ of quadratic forms in $\mathbb{R}^n$ is splitted in a direct sum $\mathcal{Q}_1\oplus...\oplus \mathcal{Q}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb{R}^n$, assume that there exist a real number $a$ such that $E(X|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal{Q}_i.$ We prove that this happens only when $k=2$, when $\mathbb{R}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure., Comment: 11 pages

Given an arbitrary three-dimensional correlation matrix, we prove that there exists a three-dimensional joint distribution for the random variable $(X,Y,Z)$ such that $X$,$Y$ and $Z$ are identically distributed with beta distribution $\beta_{k,k}(dx)$ on $(0,1)$ if $k\geq 1/2$. This implies that any correlation structure can be attained for three-dimensional copulas., Comment: 15 pages, 2 figures

In a recent paper (Asci \textit{et al.}, 2008) it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function $_{2}F_{1}$. In the present paper we fully exploit a formula due to Thomae (1879) in order to generalize substantially the class of random continuous fractions with a density of the above form. This involves the design of seven particular graphs. Infinite paths on them lead to random continued fractions with an explicit distribution. A careful study about the set of five real parameters leading to a beta-hypergeometric distribution is required, relying on almost forgotten results mainly due to Felix Klein., Comment: 28 pages, 11 figures.

Comment on ``Lancaster Probabilities and Gibbs Sampling'' [arXiv:0808.3852], Comment: Published in at http://dx.doi.org/10.1214/08-STS252A the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

When considering a graphical Gaussian model ${\mathcal{N}}_G$ Markov with respect to a decomposable graph $G$, the parameter space of interest for the precision parameter is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. The parameter space for the scale parameter of ${\mathcal{N}}_G$ is the cone $Q_G$, dual to $P_G$, of incomplete matrices with submatrices corresponding to the cliques of $G$ being positive definite. In this paper we construct on the cones $Q_G$ and $P_G$ two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior., Comment: Published at http://dx.doi.org/10.1214/009053606000001235 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)