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In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag&ndash, Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag&ndash, Leffler distributions are considered in detail. Their relations with multivariate `ordinary&rsquo, Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag&ndash, Leffler and generalized Mittag&ndash, Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value at Risk for multivariate random variables as well as stochastic orders are introduced via the set-valued approach., 30 pages, 11 figures

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 ...

Consider the evaluation of model-based functions of cumulative distribution functions that are integrals. When the cumulative distribution function does not have a tractable form but simulation of the multivariate distribution is easily feasible, we can evaluate the integral via a Monte Carlo sample, replacing the model-based distribution function by the empirical distribution function. Given a simulation sample of size N, the naive method uses $$O(N^{2})$$ comparisons to compute the empirical distribution function at all N sample vectors. To obtain faster computational speed when N needs to be large to achieve a desired accuracy, we propose methods modified from the popular merge sort and quicksort algorithms that preserve their average $$O(N\log _{2}N)$$ complexity in the bivariate case. The modified merge sort algorithm can be extended to the computation of a d-dimensional empirical distribution function at the observed values with $$O(N\log _{2}^{d-1}N)$$ complexity. Simulation studies suggest that the proposed algorithms provide substantial time savings when N is large.

In some application, quality of product or performance of a process described by some functional relationships between some variables known as multivariate linear profile in the literature. In this paper, we propose Max-MEWMA and Max-MCUSUM control charts for simultaneous monitoring of mean vector and covariance matrix in multivariate multiple linear regression profiles in Phase II. The proposed control charts also have ability to diagnose either the location or variation of the process is responsible for out-of-control signal. The performance of the proposed control charts is compared with existing method through Monte-Carlo simulations. Finally, the applicability of the proposed control charts is illustrated using a real case of calibration application in the automotive industry.

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.

The problem of testing mutual independence of p random vectors in a general setting where the dimensions of the vectors can be different and the distributions can be discrete, continuous or both is of great importance. We propose such a test which utilizes multivariate characteristic functions and is a generalization of known results. We characterize the limiting distribution of the test statistic under the null hypothesis. The limiting null distribution is approximated and the method is validated. Numerical results based on simulations are investigated and our methodology is implemented in the R package IndependenceTests. Power comparisons are also presented for some partial cases of our general test, where some competitive procedures exist.

In this paper, we first discuss some of the well-known methods available in the literature for the estimation of the parameters of a univariate/multivariate stable distribution. Based on the availa...

Skew-normal distribution is an extension of the normal distribution where the symmetry of the normal distribution is distorted with an extra parameter. A multivariate skew-normal distribution has been parametrized differently to stress different aspects and constructions behind the distribution. There are several possible parametrizations available to define the skew-normal distribution. The current most common parametrization is through Ω and α , as an alternative, parametrization through Ω and δ can be used if straightforward relation to marginal distributions is of interest. The main problem with { Ω , δ }-parametrization is that the vector δ cannot be chosen independently of Ω . This motivated us to investigate what are the possibilities of choosing δ under different correlation structures of Ω . We also show how the assumptions on structure of δ and Ω affect the asymmetry parameter α and correlation matrix R of corresponding skew-normal random variable.

We propose a multivariate functional linear regression (mFLR) approach to analysis and prediction of multivariate functional data in cases in which both the response and predictor variables contain multivariate random functions. The mFLR model, coupled with the multivariate functional principal component analysis approach, takes the advantage of cross-correlation between component functions within the multivariate response and predictor variables, respectively. The estimate of the matrix of bivariate regression functions is consistent in the sense of the multi-dimensional Gram-Schmidt norm and is asymptotically normally distributed. The prediction intervals of the multivariate random trajectories are available for predictive inference. We show the finite sample performance of mFLR by a simulation study and illustrate the method through predicting multivariate traffic flow trajectories for up-to-date and partially observed traffic streams.

In this paper, we obtain some conditions to compare multivariate mixture models with respect to some well-known multivariate stochastic orders. We also utilize the established results in reliability theory to compare the vectors of residual life-lengths of live components of ( n − k + 1 ) -out-of- n systems in both one sample and two samples situations.

Resumen El objetivo de esta investigación es proponer un modelo de volatilidad multivariable, el cual combina la propiedad de la distribución α-estable para ajustar colas pesadas con el modelo GARCH para capturar clúster de volatilidad. El supuesto inicial es que los rendimientos siguen una distribución sub-Gaussiana, la cual es un caso particular de las distribuciones estables multivariadas. El modelo GARCH propuesto se aplica en la estimación del VaR a un portafolio compuesto por cinco activos que cotizan en la Bolsa Mexicana de Valores (BMV). En particular, se compara el desempeño del modelo propuesto con la estimación del VaR obtenida bajo la hipótesis multivariada Gaussiana, t-Student y Cauchy durante el período de la crisis financiera de 2008. Abstract The purpose of this investigation is to propose a multivariate volatility model that takes into consideration time varying volatility and the property of the α-stable sub-Gaussian distribution to model heavy tails. The principal assumption is that returns follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. The proposed GARCH model is applied to a Value at Risk (VAR) estimation of a portfolio composed by 5 companies listed in the Mexican Stock Exchange Index (IPC) and compared with the one obtained using the normal multivariate distribution, t-Student and Cauchy. In particular, we examine performances during the financial crisis of 2008.

We employ a variables-in-common method for constructing multivariate Tweedie distributions, based on linear combinations of independent univariate Tweedie variables. The method lies on the convolution and scaling properties of the Tweedie laws, using the cumulant generating function for characterization of the distributions and correlation structure. The routine allows the equivalence between independence and zero correlation and gives a parametrization through given values of the mean vector and dispersion matrix, similarly to the Gaussian vector. Our approach leads to a matrix representation of multivariate Tweedie models, which permits the simulations of many known distributions, including Gaussian, Poisson, non-central gamma, gamma, and inverse Gaussian, both positively or negatively correlated.

Recently, some well-known univariate aging classes of lifetime distributions have been characterized by means of properties of their quantile functions and excess-wealth functions. The generalization of the univariate aging notions to the multivariate case involve, among other factors, appropriate definitions of multivariate quantiles or regression representation and related notions, which are able to correctly describe the intrinsic characteristic of the concepts of aging that should be generalized. The multivariate versions of these notions, which are characterized by using the multivariate u-quantiles and the multivariate excess-wealth function, are considered in this paper. Relationships between such multivariate aging classes are studied, and examples are provided.

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off betwe...

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).

In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β , multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced.

We consider break-detection procedures for vector observations, both under independence as well as under an underlying structural time series scenario. The new methods involve L2-type criteria based on empirical characteristic functions. Asymptotic as well as Monte-Carlo results are presented. The new methods are also applied to time-series data from the financial sector.

In this paper, we consider the multivariate ARMA–GARCH process governed by Markov switching coefficients. We show under proper assumptions that the process holds the L2-NED property and obeys the multivariate functional central limit theorem. The multivariate Markov switching constant conditional correlation(CCC)-GARCH model is considered as a special case.

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 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 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).

This paper proposes a new model named as the multivariate generalized Poisson log- t geometric process (MGPLTGP) model to study multivariate time-series of counts with overdispersion or underdispersion, non-monotone trends within each time-series and positive or negative correlation between pairs of time-series. This model assumes that the multivariate counts follow independent generalized Poisson distributions with an additional parameter to adjust for different degrees of dispersion including overdispersion and underdispersion. Their means after discounting the trend effect geometrically by ratio functions form latent stochastic processes and follow a multivariate log- t distribution with a flexible correlation structure to capture both positive correlation and negative correlation. By expressing the multivariate Student’s t -distribution in scale mixtures of normals, the model can be implemented through Markov chain Monte Carlo algorithms via the user-friendly WinBUGS software. The applicability of the MGPLTGP model is illustrated through an analysis of the possession and/or use of two illicit drugs, amphetamines and narcotics in New South Wales, Australia.

Multivariate outcomes with heavy skewness and thick tails often arise from clustered experiments or longitudinal studies. Linear mixed models with multivariate skew-t (MST) distributions for the random effects and the error terms is a popular tool of robust modeling for such outcomes. However the usual MST distribution only allows a common degree of freedom for all marginal distributions, which is only appropriate when each marginal has the same amount of tail heaviness. In this paper, we introduce a new class of extended MST distributions, which allow different degrees of freedom and thereby can accommodate heterogeneity in tail-heaviness across outcomes. The extended MST distributions yield a flexible family of models for multivariate outcomes. The hierarchical representation of the MST distribution allows MCMC methods to be easily applied to compute the parameter estimates. The proposed model is applied to data from two biomedical studies: one on bivariate markers of AIDS progression and the other on sexual behavior from a longitudinal study.

Distribution of the sum of symmetric lattice vectors with supports on coordinate axes is approximated by multivariate compound Poisson distribution. The characteristic function and Kerstan's methods are used to obtain local estimates and estimates in total variation.

During the last decades, no researches have conducted in order to prove some properties of the of the multivariate power series distribution, as results of the present study proved that any multivariate power series distribution is determined uniquely from the mean –function of any marginal random variable. Furthermore these results indicated also that any given function satisfying certain conditions construct a random vector with multivariate power series distribution which has a mean of the marginal random variable. A useful technique can be applied in model building when we have information about the mean- function.

Folded normal distribution arises when we try to find out the distribution of absolute values of a function of a normal variable. The properties and uses of univariate and bivariate folded normal distribution have been studied by various researchers. We study here the properties of multivariate folded normal distribution and indicate some areas of applications.

In this paper, we develop modified versions of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard adjusted likelihood ratio statistics, which follow a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed tests display superior finite sample behavior as compared to the standard likelihood ratio test. We illustrate the usefulness of our results in applied settings using a data set from the WHO MONICA Projection cardiovascular disease., Comment: 22 pages, 3 figures

Birnbaum and Saunders introduced in 1969 a two-parameter lifetime distribution which has been used quite successfully to model a wide variety of univariate positively skewed data. Diaz-Garcia and Leiva-Sanchez [8] proposed a generalized Birnbaum-Saunders distribution by using an elliptically symmetric distribution in place of the normal distribution. Recently, Kundu et al. [13] introduced a bivariate Birnbaum-Saunders distribution, based on a transformation of a bivariate normal distribution, and discussed its properties and associated inferential issues. In this paper, we construct a generalized multivariate Birnbaum-Saunders distribution, by using the multivariate elliptically symmetric distribution as a base kernel for the transformation instead of the multivariate normal distribution. Different properties of this distribution are obtained in the general case. Special emphasis is placed on statistical inference for two particular cases: (i) multivariate normal kernel and (ii) multivariate-t kernels. We use the maximized log-likelihood values for selecting the best kernel function. Finally, a data analysis is presented for illustrative purposes.

In the treatment of the sequential beta-protection procedure, we define the reasonable stopping time and investigate that for the stopping time Wijsman`s requirements, coverage probability and beta-protection conditions, are satisfied in the estimation for the mean vector by the sample from the multivariate normal distributed population with unknown mean vector and a positive definite variance-covariance matrix .

Multivariate meta-analysis is becoming more commonly used. Methods for fitting the multivariate random effects model include maximum likelihood, restricted maximum likelihood, Bayesian estimation and multivariate generalisations of the standard univariate method of moments. Here, we provide a new multivariate method of moments for estimating the between-study covariance matrix with the properties that (1) it allows for either complete or incomplete outcomes and (2) it allows for covariates through meta-regression. Further, for complete data, it is invariant to linear transformations. Our method reduces to the usual univariate method of moments, proposed by DerSimonian and Laird, in a single dimension. We illustrate our method and compare it with some of the alternatives using a simulation study and a real example.

International audience; In the context of multivariate multilevel data analysis, this paper focuses on the multivariate linear mixed-effects model, including all the correlations between the random effects when the dimensional residual terms are assumed uncorrelated. Using the EM algorithm, we suggest more general expressions of the model's parameters estimators. These estimators can be used in the framework of the multivariate longitudinal data analysis as well as in the more general context of the analysis of multivariate multilevel data. By using a likelihood ratio test, we test the significance of the correlations between the random effects of two dependent variables of the model, in order to investigate whether or not it is useful to model these dependent variables jointly. Simulation studies are done to assess both the parameter recovery performance of the EM estimators and the power of the test. Using two empirical data sets which are of longitudinal multivariate type and multivariate multilevel type, respectively, the usefulness of the test is illustrated.

As alternatives to the normal distributions, $t$ distributions are widely applied in robust analysis for data with outliers or heavy tails. The properties of the multivariate $t$ distribution are well documented in Kotz and Nadarajah's book, which, however, states a wrong conclusion about the conditional distribution of the multivariate $t$ distribution. Previous literature has recognized that the conditional distribution of the multivariate $t$ distribution also follows the multivariate $t$ distribution. We provide an intuitive proof without directly manipulating the complicated density function of the multivariate $t$ distribution.

We give multivariate generalization of Macdonald distribution and study several of its properties. We also define the multivariate Macdonald-gamma distribution and derive a number of results pertaining to it.

In the analysis procedures introduced in the last four chapters, parameters are estimated by the least squares (LS) method, as reviewed in Sect. 8.1. The remaining sections in this chapter serve to prepare readers for the following chapters, in which a maximum likelihood (ML) method, which differs from LS, is used for estimating parameters. That is, the ML method is introduced in Sect. 8.2, which is followed by describing the notion of probability density function and the ML method with multivariate normal distribution. Finally, ML-based model selection with information criteria is introduced.

We investigate the finite-sample properties of certain procedures which employ the novel notion of the probability weighted empirical characteristic function. The procedures considered are: (1) Testing for symmetry in regression, (2) Testing for multivariate normality with independent observations, and (3) Testing for multivariate normality of random effects in mixed models. Along with the new tests alternative methods based on the ordinary empirical characteristic function as well as other more well known procedures are implemented for the purpose of comparison.