Structural decompositions of multivariate distributions with applications in moment and cumulant

We provide lattice decompositions for multivariate distributions. The lattice decompositions reveal the structural relationship between the Lancaster/Bahadur model and the model of Streitberg (Ann. Statist. 18 (1990) 1878). For multivariate categorical data, the decompositions allows modeling strategy for marginal inference. The theory discussed in this paper illustrates the concept of reproducibility, which was discussed in Liang et al. (J. Roy. Statist. Soc. Ser. B 54 (1992) 3). For the purpose of delineating the relationship between the various types of decompositions of distributions, we develop a theory of polytypefication, the generality of which is exploited to prove results beyond interaction.     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

Singular random matrix decompositions: distributions

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions

We consider here the distributions of order statistics and linear combinations of order statistics from an elliptical distribution. We show that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture representations to derive their moment generating functions and moments, when they exist.     

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.     

Flexible bivariate beta distributions

Bivariate beta distributions which can be used to model data sets exhibiting positive or negative correlation are introduced. Properties of these bivariate beta distributions and their applications in Bayesian analysis are discussed. Three methods for parameter estimation are presented. The performance of these estimators is evaluated based on Monte Carlo simulations. Examples are provided to illustrate how additional parameters can be introduced to gain even more modeling flexibility. A possible extension of the proposed bivariate beta model and a multivariate generalization are also discussed.     

Wishart distributions associated with matrix quadratic forms

For a normally distributed random matrix Y with mean zero and general covariance matrix Σ_Y and for a symmetric matrix W, necessary and sufficient conditions are derived for the Wishartness of Y′WY.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

On the minimization of multinomial tails and the Gupta-Nagel conjecture

This paper is primarily concerned with the open problem of minimizing the lower tail of the multinomial distribution. During the study of that specific problem, we have developed an approach which reveals itself useful for solving a general class of problems involving multinomial probabilities. Concerning the main problem, we provide a self-contained proof that the minimum of the multinomial lower tail is reached, as conjectured by Gupta and Nagel (Sankhya Ser. B 29 (1967) 1) (within the framework of subset-selection problems) at the equal probability configuration, i.e., when the cell probabilities are equal to one another. We also point out some novel inequalities and general properties involving multinomial probabilities and multinomial coefficients.     

Kronecker product permutation matrices and their application to moment matrices of the normal distribution

In this paper, we consider the matrix which transforms a Kronecker product of vectors into the average of all vectors obtained by permuting the vectors involved in the Kronecker product. An explicit expression is given for this matrix, and some of its properties are derived. It is shown that this matrix is particularly useful in obtaining compact expressions for the moment matrices of the normal distribution. The utility of these expressions is illustrated through some examples.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.     

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.     

Wishart-Laplace distributions associated with matrix quadratic forms

For a normal random matrix Y with mean zero, necessary and sufficient conditions are obtained for Y^′W_kY to be Wishart-Laplace distributed and {Y^′W_kY} to be independent, where each W_k is assumed to be symmetric rather than nonnegative definite.     

An extension of the factorization theorem to the non-positive case

This paper presents a method of determining joint distributions by known conditional distributions. A generalization of the Factorization Theorem is proposed. The generalized theorem is proved under the assumption that the support of unknown joint distribution may be divided into a countable number of sets, which all satisfy the relative weak positivity condition. This condition is defined in the paper and it generalizes the positivity condition introduced by Hammersley and Clifford. The theorem is illustrated with three examples. In the first example we determine a joint density in the case when the support of an unknown density is a continuous nonproduct set from Euclidean space . In the second example we seek the joint probability for the number of trials and the number of successes in Bernoulli's scheme. We also examine a simple example given by Kaiser and Cressie (J. Multivariate Anal. 73 (2000) 199).     

Stein's Lemma for elliptical random vectors

For the family of multivariate normal distribution functions, Stein's Lemma presents a useful tool for calculating covariances between functions of the component random variables. Motivated by applications to corporate finance, we prove a generalization of Stein's Lemma to the family of elliptical distributions.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.