Class Of Multivariate Distributions Research Articles

Multivariate zero-inflated Bell distribution and its inference and applications

In this paper we introduce a multivariate family of distributions for multivariate count data with excess zeros, which is a multivariate extension of the univariate zero-inflated Bell distribution. We derive various general properties of this multivariate distribution. In particular, the marginal distributions are univariate zero-inflated Bell distributions. The model parameters are estimated using the traditional maximum likelihood estimation method. In addition, we develop a simple EM algorithm to compute the maximum likelihood estimates of the parameters of the new multivariate distribution with closed-form expressions for the maximum likelihood estimators. Empirical applications that employ real multivariate count data are considered to illustrate the usefulness of the new class of multivariate distributions, and comparisons with the multivariate zero-inflated Poisson distribution, multivariate zero-adjusted Poisson distributions, and multivariate zero-inflated generalized Poisson distribution are made.

The Sibling Distribution for Multivariate Life Time Data

AbstractA flexible class of multivariate distributions for continuous lifetimes is proposed. The distribution is defined in terms of the age-at-death of m siblings. The expression for the joint density is derived using classical results from mathematical demography. The parameters of the distribution are the age-specific birth and death rates, in addition to a vector of relative death times for the m siblings. For the case of constant birth and death rates we are able to derive an explicit expression for the bivariate sibling density, which is proven to be MTP2, and hence has positive dependence. Further, we show that a special case of the sibling distribution belongs to the Block-Basu class of multivariate distribution. In the general case, with age-dependent birth and death rates, evaluation of the density involves numerical integration, but is still feasible.

High-dimensional consistent independence testing with maxima of rank correlations

Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting nonlinear, nonmonotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence. Built upon a newly developed Cramér-type moderate deviation theorem for degenerate U-statistics, our results cover a variety of rank correlations including Hoeffding’s $D$, Blum–Kiefer–Rosenblatt’s $R$ and Bergsma–Dassios–Yanagimoto’s $\tau^{*}$. The proposed tests are distribution-free in the class of multivariate distributions with continuous margins, implementable without the need for permutation, and are shown to be rate-optimal against sparse alternatives under the Gaussian copula model. As a by-product of the study, we reveal an identity between the aforementioned three rank correlation statistics, and hence make a step towards proving a conjecture of Bergsma and Dassios.

Grouped Normal Variance Mixtures

Grouped normal variance mixtures are a class of multivariate distributions that generalize classical normal variance mixtures such as the multivariate t distribution, by allowing different groups to have different (comonotone) mixing distributions. This allows one to better model risk factors where components within a group are of similar type, but where different groups have components of quite different type. This paper provides an encompassing body of algorithms to address the computational challenges when working with this class of distributions. In particular, the distribution function and copula are estimated efficiently using randomized quasi-Monte Carlo (RQMC) algorithms. We propose to estimate the log-density function, which is in general not available in closed form, using an adaptive RQMC scheme. This, in turn, gives rise to a likelihood-based fitting procedure to jointly estimate the parameters of a grouped normal mixture copula jointly. We also provide mathematical expressions and methods to compute Kendall’s tau, Spearman’s rho and the tail dependence coefficient λ. All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).

Multivariate Fractional Phase—Type Distributions

Abstract We extend the Kulkarni class of multivariate phase–type distributions in a natural time–fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non–Markovian jump process with ML sojourn times. This new class complements an earlier multivariate ML construction [2] and in contrast to the former also allows for tail dependence. We derive properties and characterizations of this class, and work out some special cases that lead to explicit density representations.

Maximum likelihood estimation for totally positive log‐concave densities

AbstractWe study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log‐supermodular (MTP2) distributions and log‐L♮‐concave (LLC) distributions. In both cases we also assume log‐concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}d or in under MTP2, and for samples in under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.

Concentration of kernel matrices with application to kernel spectral clustering

We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on $\mathbb{R}^{d}$, including the strongly log-concave distributions under affine transformations. A feature of our result is that the data points need not have identical distributions or zero mean, which is key in certain applications such as clustering. Our bound for the Lipschitz kernels is dimension-free and sharp up to constants. For comparison, we also derive the companion result for the Euclidean (inner product) kernel for a class of sub-Gaussian distributions. A notable difference between the two cases is that, in contrast to the Euclidean kernel, in the Lipschitz case, the concentration inequality does not depend on the mean of the underlying vectors. As an application of these inequalities, we derive a bound on the misclassification rate of a kernel spectral clustering (KSC) algorithm, under a perturbed nonparametric mixture model. We show an example where this bound establishes the high-dimensional consistency (as $d\to \infty $) of the KSC, when applied with a Gaussian kernel, to a noisy model of nested nonlinear manifolds.

New Results on Truncated Elliptical Distributions

Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions. Two members of this class are the multivariate truncated normal and multivariate truncated t distributions. We derive statistical properties of the truncated elliptical distributions. Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions.

Characterization and classification of multiple stable Tweedie models

Extending normal stable Tweedie models, the multiple stable Tweedie (MST) models are recently introduced as a huge class of multivariate distributions. They are composed by a fixed univariate stable Tweedie variable having a positive mean domain and random variables that, given the fixed one, are real independent stable Tweedie variables, possibly different, with the same dispersion parameter equal to the fixed component.Within the framework of exponential dispersion models, we completely prove the characterization of the MST models through their variance functions under steepness property. Thereforewe deduce a new classification of the Poisson-MST, gamma-MST, noncentral-gamma-MST, and inverse-Gaussian-MST families, where each of them contains one element of the normal stable Tweedie models, namely normal-Poisson, normal-gamma (or gamma-Gaussian), normal-noncentral-gamma, and normal-inverse-Gaussian distributions, respectively.

Simulation of some multivariate distributions related to the Dirichlet distribution with application to Monte Carlo simulations

ABSTRACTSimulation of multivariate distributions is important in many applications but remains computationally challenging in practice. In this article, we introduce three classes of multivariate distributions from which simulation can be conducted by means of their stochastic representations related to the Dirichlet random vector. More emphasis is made to simulation from the class of uniform distributions over a polyhedron, which is useful for solving some constrained optimization problems and ha`s many applications in sampling and Monte Carlo simulations. Numerical evidences show that, by utilizing state-of-the-art Dirichlet generation algorithms, the introduced methods become superior to other approaches in terms of computational efficiency.

Variance-mean mixture of the multivariate skew normal distribution

In this paper, we introduce a new class of multivariate distributions as an extension of the normal variance–mean mixture distributions class. The new class results from a variance-mean mixture of the skew normal and the generalized inverse Gaussian distributions. The new class is very flexible in terms of heavy tails and skewness and many of the widely used distributions, such as generalized hyperbolic, skew t, and skew Laplace distributions are included as special or limiting cases of the new class. An explicit expression for the density function of the new class is given and some of its distributional properties, such as moment generating function, linear transformations, quadratic forms, marginal and conditional distributions are examined. We give a simulation algorithm to generate random variates from the new class and propose an EM algorithm for maximum likelihood estimation of its parameters. We provide some examples to demonstrate the modeling strength of the proposed class.

Bayesian prediction under a class of multivariate distributions

In this paper the prediction problem is studied under members of a class \({\Im^{*}}\) of multivariate distributions, constructed by AL-Hussaini and Ateya (Stat Pap 46:321–338, 2005; J Egypt Math Soc 14(1):45–54, 2006). More attention is given to bivariate compound Rayleigh distribution, which is a member of this class, as illustrative example.

Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences

This paper deals with a surprising connection between exchangeable distributions on {0,1}n and the recently introduced Levy-frailty copulas, the link being provided by a new class of multivariate distribution functions called linearly order symmetric. The characterisation theorem for Levy-frailty copulas is given a new and short (non-combinatorial) proof, and a related result is shown for exchangeable Marshall–Olkin distributions. A common thread in all these considerations is higher-order monotonic functions on integer intervals of the form {0,1,…,n}.

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.

Extending the multivariate generalised [formula omitted] and generalised [formula omitted] distributions

The G G H family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting G G H family encompasses the multivariate generalised hyperbolic ( G H ), which itself contains the multivariate t and multivariate Variance-Gamma ( V G ) distributions as special cases. It also contains the generalised multivariate t distribution [O. Arslan, Family of multivariate generalised t distribution, Journal of Multivariate Analysis 89 (2004) 329–337] and a new generalisation of the V G as special cases. Our approach unifies into a single G H -type family the hitherto separately treated t -type [O. Arslan, A new class of multivariate distribution: Scale mixture of Kotz-type distributions, Statistics and Probability Letters 75 (2005) 18–28; O. Arslan, Variance–mean mixture of Kotz-type distributions, Communications in Statistics-Theory and Methods 38 (2009) 272–284] and V G -type cases. The G G H distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate V G [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467–1475] does however extend in one sense to their generalisations.

On the existence of maximum likelihood estimates in random effects models for clustered multivariate binary data

We consider a class of random effects models for clustered multivariate binary data based on the threshold crossing technique of a latent random vector. Components of this latent vector are assumed to have a Laird–Ware structure. However, in place of their Gaussian assumptions, any specified class of multivariate distribution is allowed for the random effects, and the error vector is allowed to have any strictly positive pdf. A well known member of this class of models is the multivariate probit model with random effects. We investigate sufficient and necessary conditions for the existence of maximum likelihood estimates for the location and the association parameters. Implications of our results are illustrated through some hypothetical examples.

Compound and scale mixture of matricvariate and matrix variate Kotz-type distributions

We have derived the generalised Wishart and inverse-Wishart distributions based on the matrix variate and matricvariate Kotz-type distributions. The scale mixture of matricvariate and matrix variate Kotz-type distributions and the inverse generalised gamma distribution are then proposed. It is shown that the class of distributions termed the family of t -type distributions proposed by Arslan [Arslan, O. (2005). A new class of multivariate distributions: Scale mixture of Kotz-type distributions. Statistics and Probability Letters, 76, 18–28] is obtained as particular cases of the result given here. We have also derived the compound matricvariate and matrix variate Kotz-type distributions and the inverse generalised Wishart based on the matrix variate and matricvariate Kotz-type distributions. These latter results lead us to propose a different matricvariate version of the family of t -type distributions and others cited in the above-mentioned reference.

A general class of multivariate distribution involving H̅ -function

In this paper an attempt has been made to present unified theory of the classical statistical distribution associated with the multivariate generalized Dirichlet distribution involving H- function with general arguments. In particular, Mathematical expectation of a general class of polynomials, characteristic function and the distribution function are investigated.

A general class of multivariate distribution involving H –function

A general class of multivariate distribution involving H –function

Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers

We consider classes of multivariate distributions which can model skewness and are closed under orthogonal transformations. We review two classes of such distributions proposed in the literature and focus our attention on a particular, yet quite flexible, subclass of one of these classes. Members of this subclass are defined by affine transformations of univariate (skewed) distributions that ensure the existence of a set of coordinate axes along which there is independence and the marginals are known analytically. The choice of an appropriate m-dimensional skewed distribution is then restricted to the simpler problem of choosing m univariate skewed distributions. We introduce a Bayesian model comparison setup for selection of these univariate skewed distributions. The analysis does not rely on the existence of moments (allowing for any tail behaviour) and uses equivalent priors on the common characteristics of the different models. Finally, we apply this framework to multi-output stochastic frontiers using data from Dutch dairy farms.