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In this chapter we extend the theory from univariate distributions to multivariate formulations for a select few distributions: the Gaussian, lognormal, and gamma distributions. We shall also introduce mixed distribution, which is a combination of Gaussian and lognormal random variables. We shall highlight that not all of the desired properties of the univariate descriptive statistics carry over to the multivariate formulation, but start to lay the foundations for the derivation of many of the data assimilation schemes in later chapters.

We present a hierarchical decomposition scheme for computing the n-dimensional integral of multivariate normal probabilities that appear frequently in statistics. The scheme exploits the fact that ...

In this paper, we consider Sarmanov’s multivariate discrete distribution as counting distribution in two multivariate compound models: the first model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov’s distribution can join different types of marginals, we also assume that these marginals belong to Panjer’s class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

In this paper, asymptotic normality is established for the parameters of the multivariate skew-normal distribution under two parametrizations. Also, an analytic expression and an asymptotic...

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.

Most existing control charts monitoring the covariance matrix of multiple variables were restricted to multivariate normal distribution. When the process distribution is non-normal, the performance...

A multivariate probability distribution model for seven parameters of coarse-grained soils is constructed based on the SAND/7/2794 database that was compiled by the authors. It is shown that the multivariate probability distribution captures the correlation behaviors in the database among the seven parameters. This multivariate distribution model serves as a prior distribution model in the Bayesian analysis and can be updated into the posterior distribution of the design soil parameter when multivariate site-specific information is available. It is shown that this Bayesian analysis is conceptually similar to what is routinely carried out in practice, which utilizes information from comparable sites to supplement limited site-specific information. The resulting posterior distribution from Bayesian analysis merely combines different uncertainties associated with different sources of “correlated” information in a more consistent way. In this paper, the parameters for the posterior distribution of the design soil parameter are summarized into engineer-friendly tables ( Tables 9 and 10 ) so that engineers do not need to conduct the actual Bayesian analysis. Caution should be taken in extrapolating the results of this paper to cases that are not covered by SAND/7/2794, because the resulting posterior distribution can be misleading. This caveat applies to conventional regression equations as well.

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial distribution, and some discrete distributions arising from the multivariate discrete normal distribution.

This article introduces a new asymmetric distribution constructed by assuming the multivariate normal mean-variance mixture model. Called normal mean-variance mixture of the Lindley distribution, w...

The multivariate log-normal distribution is used by many authors and statistical uncertainty propagation programs for inherently positive quantities. Sometimes it is claimed that the log-normal distribution results from the maximum entropy principle, if only means, covariances and inherent positiveness of quantities are known or assumed to be known. In this article we show that this is not true. Assuming a constant prior distribution, the maximum entropy distribution is in fact a truncated multivariate normal distribution – whenever it exists. However, its practical application to multidimensional cases is hindered by lack of a method to compute its location and scale parameters from means and covariances. Therefore, regardless of its theoretical disadvantage, use of other distributions seems to be a practical necessity.

A multivariate semi-α-Laplace distribution (denoted by Ms-αLaplace) is introduced and studied in this paper. It is more general than the multivariate Linnik and Laplace distributions proposed by Sabu and Pillai (1991) or Anderson (1992). The Ms-αLaplace distribution has univariate semi-α-Laplace (Pillai, 1985) as marginal distribution. Various characterization theorems of the Ms-αLaplace distribution based on the closure property of the normalized geometric sum are proved.

One approach to smoothing empirical score distributions is to apply a probability distribution. However, the score distributions sometimes form a ‘toothed’ shape, meaning that the probability distribution does not always fit well. The normal distribution can be applied for the fitting; however, this has a symmetric shape and a domain ranging from minus infinity to plus infinity. Polynomial log-linear models are often used for fitting frequency distributions, especially those with complicated shapes, but these do not provide information on the frequency distribution. In this study, we propose the maximum likelihood model for approximating a discrete distribution. Our approach is based on the mixed-multivariate beta distribution, which can cover asymmetric score distributions with known upper and lower bounds. Further, a simulation study is conducted to compare the results from the application of the mixed-multivariate beta distribution, mixed multivariate normal distribution, and two polynomial log-linear models. Results from a real data analysis are also given in this research.

In this paper, we generalize the Wishart distribution utilizing a fresh approach that leads to the hypergeometric Wishart generator distribution with the Wishart generator and the Wishart as special cases. Important statistical characteristics are derived. The significance of this generator distribution is further demonstrated by assuming a special case as a prior for the underlying matrix variate normal model.

In this article we obtain the exact joint distribution of a ranked set sample. We show that this distribution belongs to the family of unified multivariate skew normal distributions. We also investigate a multivariate skew-t distribution 62J12 using ranked set samples as an application of our results. A numerical example is also provided to illustrate our results.

This article proposes the elliptical multivariate leptokurtic-normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram–Charlier expansion of the multivariate normal (MN) distribution and has a closed form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model-based clustering. An EM algorithm is presented to specifically obtain maximum likelihood estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions. The Canadian Journal of Statistics xx: 1–25; 2016 © 2016 Statistical Society of Canada

In this article, by considering a multivariate normal mean–variance mixture distribution, we derive the exact joint distribution of linear combinations of order statistics and their concomitants. From this general result, we then deduce the exact marginal and conditional distributions of order statistics and their concomitants arising from this distribution. We finally illustrate the usefulness of these results by using a Swiss markets dataset.

We study some of the main probabilistic properties of the so called multivariate skew-normal-Cauchy distribution. Simple expressions to compute the entries of the expected Fisher information matrix of this multivariate distribution are proposed.

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component-wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max-stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.

Multivariate t mixture (TMIX) models have emerged as a powerful tool for robust modeling and clustering of heterogeneous continuous multivariate data with observations containing longer than normal tails or atypical observations. In this paper, we explicitly derive the score vector and Hessian matrix of TMIX models to approximate the information matrix under the general and three special cases. As a result, the standard errors of maximum likelihood (ML) estimators are calculated using the outer-score, Hessian matrix, and sandwich-type methods. We have also established some asymptotic properties under certain regularity conditions. The utility of the new theory is illustrated with the analysis of real and simulated dataźsets.

Univariate Birnbaum-Saunders distribution has received a considerable attention in recent years. Rieck and Nedelman (Technometrics, vol. 33, 51–60, 1991) introduced a log Birnbaum-Saunders distribution. We introduce a multivariate log Birnbaum-Saunders distribution and discuss its different properties. It is observed that the proposed multivariate model can be obtained from the multivariate Gaussian copula. We have proposed the maximum likelihood estimators of the unknown parameters. Since it is a computationally challenging problem, particularly if the dimension is high, we have considered the approximate maximum likelihood estimators based on the Copula structure using two-step procedure. The asymptotic distributions of both these estimators have been obtained. We compare their performances using Monte Carlo simulations, and it is observed that their performances are very similar in nature. One data set has been analyzed for illustrative purposes.

In this article, we develop an algorithm for working out the distribution of proposed test statistics. The computed distribution is compared with its existing asymptotic normal approximation using convergence rates and disparity measures. Simulation study is made to estimate the level of significance, and power to assess the performance of algorithm based procedure in comparison to asymptotic normal approximation. Both power comparisons and convergence rates suggest that the distribution computed through algorithm procedure be used for smaller sample sizes.

This work treats non-parametric estimation of multivariate probability mass functions, using multivariate discrete associated kernels. We propose a Bayesian local approach to select the matrix of bandwidths considering the multivariate Dirac Discrete Uniform and the product of binomial kernels, and treating the bandwidths as a diagonal matrix of parameters with some prior distribution. The performances of this approach and the cross-validation method are compared using simulations and real count data sets. The obtained results show that the Bayes local method performs better than cross-validation in terms of integrated squared error.

Longitudinal studies often involve multiple outcomes measured repeatedly from the same subject. The analysis of multivariate longitudinal data can be challenging due to its complex correlated nature. In this paper, we develop multivariate marginal models in longitudinal studies with multiple response variables, and improve parameter estimation by incorporating informative correlation structures. In theory, we show that the proposed method yields a consistent and efficient estimator which follows an asymptotic normal distribution. Monte Carlo studies indicate that the proposed method performs well in the sense of reducing bias and improving estimation efficiency. In addition, the proposed approach is applied to a real longitudinal data example of transportation safety with different response families.

Random vectors with positive components are common in many applied fields, for example, in meteorology, when daily precipitation is measured through a region Marchenko and Genton (2010). Frequently, the log-normal multivariate distribution is used for modeling this type of data. This modeling approach is not appropriate for data with high asymmetry or kurtosis. Consequently, more flexible multivariate distributions than the log-normal multivariate are required. As an alternative to this distribution, we propose the log-alpha-power multivariate and log-skew-normal multivariate models. The first model is an extension for positive data of the fractional order statistics model Durrans (1992). The second one is an extension of the log-skew-normal model studied by Mateu-Figueras and Pawlowsky-Glahn (2007). We study parameter estimation for these models by means of pseudo-likelihood and maximum likelihood methods. We illustrate the proposal analyzing a real dataset.

We present in this note a sufficient condition for sums of TP 2 functions (totally positive of order 2) to be TP 2 . We apply our results to several distributions and provide, in particular, conditions for which certain widely used families of distributions are multivariate totally positive of order 2.

We introduce a new method for sampling from the Wishart distribution by representing the Wishart distributed random matrix as a function of independent multivariate normal-gamma random vectors. An efficient monotone data augmentation (MDA) algorithm is developed for Bayesian multivariate linear regression. For longitudinal outcomes, the proposed method is easier to implement and interpret than that based on Bartlett’s decomposition. The proposed algorithm is illustrated by the analysis of an antidepressant trial.

Based on geometric invariance properties, we derive an explicit prior distribution for the parameters of multivariate linear regression problems in the absence of further prior information. The problem is formulated as a rotationally-invariant distribution of \(L\)-dimensional hyperplanes in \(N\) dimensions, and the associated system of partial differential equations is solved. The derived prior distribution generalizes the already known special cases, e.g., 2D plane in three dimensions.

In this paper, we use joint and conditional distributions of order statistics from a trivariate normal distribution to obtain closed expressions for reliability measures such as mean residual lifetime for systems with three dependent components with normal lifetimes.

For the multivariate elliptical model subjective Bayesian estimators of the location vector and some functions of the characteristic matrix with the normal-inverse Wishart and the normal-Wishart as prior, respectively, are derived. Fang and Li (1999) considered the elliptical model for Bayesian analysis for an objective prior structure. In addition, the newly developed results are applied to the multivariate normal- and t-distribution. A performance study is done to evaluate the normal-gamma and normal-inverse gamma distributions as suitable priors. A practical application for the posterior distributions of the multivariate t-distribution is included by means of Gibbs sampling and a Metropolis-Hastings algorithm.

Mixtures of normal distributions provide a useful modeling extension of the normal distribution—both univariate and multivariate. Unlike the normal distribution, mixtures of normals can capture the kurtosis (fat tails) and nonzero skewness often necessary for accurately modeling a variety of real-world variables. An efficient analytical Monte Carlo method is proposed for considering multivariate mixtures of normal distributions having arbitrary covariance matrices. The method consists of a linear transformation of a multivariate normal having a computed covariance matrix into the desired multivariate mixture of normal distributions. The computed covariance matrix is derived analytically. Among the properties of the multivariate mixture of normals that we demonstrate is that any linear combination of mixtures of normal distributions is also a mixture of normal distributions. Methods of fitting mixtures of normal distributions are briefly discussed. A motivating example carried throughout this paper is the use of multivariate mixtures of normals for modeling daily changes in market variables.

Motivated from the stochastic representation of the univariate zero-inflated Poisson (ZIP) random variable, the authors propose a multivariate ZIP distribution, called as Type I multivariate ZIP distribution, to model correlated multivariate count data with extra zeros. The distributional theory and associated properties are developed. Maximum likelihood estimates for parameters of interest are obtained by Fisher's scoring algorithm and the expectation-maximization (EM) algorithm, respectively. Asymptotic and bootstrap confidence intervals of parameters are provided. Likelihood ratio test and score test are derived and are compared via simulation studies. Bayesian methods are also presented if prior information on parameters is available. Two real data sets are used to illustrate the proposed methods. Under both AIC and BIC, our analysis of the two data sets supports the Type I multivariate zero-inflated Poisson model as a much less complex alternative with feasibility to the existing multivariate ZIP models proposed by Li et?al. (Technometrics, 29-38, Vol 41, 1999).

The multivariate normal distribution was briefly introduced in Chapter 2. In this chapter we consider its properties in some detail since the estimation of its parameters is the source of many standard multivariate statistical methods. In addition, we shall meet a number of derived distributions of fundamental importance.

Sampling from a truncated multivariate distribution subject to multiple linear inequality constraints is a recurring problem in many areas in statistics and econometrics, such as the order-restricted regressions, censored data models, and shape-restricted nonparametric regressions. However, the sampling problem remains nontrivial due to the analytically intractable normalizing constant of the truncated multivariate distribution. We first develop an efficient rejection sampling method for the truncated univariate normal distribution, and analytically establish its superiority in terms of acceptance rates compared to some of the popular existing methods. We then extend our methodology to obtain samples from a truncated multivariate normal distribution subject to convex polytope restriction regions. Finally, we generalize the sampling method to truncated scale mixtures of multivariate normal distributions. Empirical results are presented to illustrate the superior performance of our proposed Gibbs sampler in...

The bivariate generalization of the Gauss hypergeometric distribution is defined by the probability density function proportional to x α1−1y α2−1 (1 − x − y) β−1 (1 + ξ1x + ξ2y) −γ , x > 0, y > 0, x + y < 1, where αi > 0, i = 1, 2, β > 0, −∞ < γ < ∞ and ξi > −1, i = 1, 2 are constants. In this article, we study several of its properties such as marginal and conditional distributions, joint moments and the coefficient of correlation. We compute the exact forms of R´enyi and Shannon entropies for this distribution. We also derive the distributions of X+Y , X/(X +Y ), V = X/Y and XY where X and Y follow a bivariate Gauss hypergeometric distribution. COL0000532

In this note we address the problem of finding the direction with maximal skewness for a random vector that follows a multivariate skew-t distribution. The result relies on the well-known representation of the multivariate skew-t distribution as a scale mixture of skew-normal multivariate distributions.

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.

Summary In this paper, we consider some results on distribution theory of multivariate progressively Type-II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property.

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

We give the cumulative distribution function (cdf) of M n , the (element-wise) maximum of a sequence of n observations from a multivariate AR ( p ) process. We do the same for a multivariate MA ( p ) process. Solutions are first given in terms of repeated integrals and then for the case, where the marginal cdf of the observations is absolutely continuous. The cdf of the multivariate maximum M n is then given as a weighted sum of the n th powers of the eigenvalues of a non-symmetric Fredholm kernel. The weights are given in terms of the left and right eigenfunctions of the kernel.

This paper proposed an alternate generalization of Sam-Solai's Multivariate additive Uniform distribution from the continuous univariate distribution. Further,we find its Marginal, Multivariate Conditional distributions, Multivariate Generating functions, Multivariate survival, hazard functions and also discussed it's special cases. The special cases include the transformation of Sam-Solai's Multivariate additive Uniform distribution into Multivariate standard additive uniform distribution and Multivariate exponential distribution. Moreover, the bivariate correlation between any two standard uniform random variables was found to be approximately -0.33.

Several fascinating examples of non-Gaussian bivariate distributions which have marginal distribution functions to be Gaussian have been proposed in the literature. These examples often clarify several properties associated with the normal distribution. In this paper, we generalize this result in the sense that we construct a p -dimensional distribution for which any proper subset of its components has the Gaussian distribution. However, the joint p -dimensional distribution is inconsistent with the distribution of these subsets because it is not Gaussian. We study the probabilistic properties of this non-Gaussian multivariate distribution in detail. Interestingly, several popular tests of multivariate normality fail to identify this p -dimensional distribution as non-Gaussian. We further extend our construction to a class of elliptically contoured distributions as well as skewed distributions arising from selections, for instance the multivariate skew-normal distribution.

In this article, a new form of multivariate slash distribution is introduced and some statistical properties are derived. In order to illustrate the advantage of this distribution over the existing generalized multivariate slash distribution in the literature, it is applied to a real data set.

In this paper we have derived the distribution of multivariate order statistics for multivariate exponential & multivariate weibull distribution. The moment expression for multivariate order statistics has also been derived.

In this article, we introduce a new three-parameter skewed distribution of which normal distribution is a special case. This distribution is obtained by using geometric sum of independent identically distributed normal random variables. We call this distribution as the geometric skew normal distribution. Different properties of this new distribution have been investigated. The probability density function of geometric skew normal distribution can be unimodal or multimodal, and it always has an increasing hazard rate function. It is an infinite divisible distribution, and it can have heavier tails. The maximum likelihood estimators cannot be obtained in explicit forms. The EM algorithm has been proposed to compute the maximum likelihood estimators of the unknown parameters. One data analysis has been performed for illustrative purposes. We further consider multivariate geometric skew normal distribution and explore its different properties. The proposed multivariate model induces a multivariate Levy process, and some properties of this multivariate process have been investigated. Finally, we conclude the paper.

In this paper we first introduce a new class of multivariate generalized asymmetric skew-normal distributions with two parameters λ1,λ2 that we present it by BGSNn, m (λ1,λ2), and we finally obtain some special properties of BGSNnm(λ1,λ2).