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The article proposes a Bayesian methodology to implement complete and cohesive analysis of vector time series. Assuming the series are generated by vector autoregressive moving average process, the...

Multivariate time series may be found in many fields of application such as economics, meteorology, and utilities. In economics, for example, one may record yearly money supply and real interest ra...

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 ...

Various methods have been proposed to monitor changes in a process covariance matrix. In view that a covariance matrix can be fully defined by its eigenvalues and eigenvectors, this paper suggests monitoring the covariance matrix based on eigenvalues as another alternative. Although there are some recent discussions about the use of eigenvalues for hypothesis testing in multivariate analysis, the use of them for monitoring covariance matrix changes has been less studied in multivariate quality control. The simulation results show that the proposed method performs especially well under simultaneous shifts in both variance and correlation elements and competitively under shifts in variance or correlation elements only, compared to the existing approaches. This demonstrates a good property of the proposed method being able to provide a robust detection performance under a wide variety of scenarios. A real example is also provided to illustrate the implementation of the proposed method.

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.

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.

Estimation in the multivariate context when the number of observations available is less than the number of variables is a classical theoretical problem. In order to ensure estimability, one has to assume certain constraints on the parameters. A method for maximum likelihood estimation under constraints is proposed to solve this problem. Even in the extreme case where only a single multivariate observation is available, this may provide a feasible solution. It simultaneously provides a simple, straightforward methodology to allow for specific structures within and between covariance matrices of several populations. This methodology yields exact maximum likelihood estimates.

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...

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...

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.

The eigenvalue distributions of complex Wishart matrices are critical research issues in random matrix theory (RMT). The scaled eigenvalue (SE) distributions of complex Wishart matrices with finite dimensions are deduced in this paper. The probability density function (PDF) and cumulative distribution function (CDF) of the SE are formulated in the closed-form and coefficient-based expressions. Moreover, the derivative of SE PDF is provided in an exact formulation utilizing the same coefficient vectors. The numerical results verify that the newly proposed SE distributions fit the empirical distributions very well and the dimensions of Wishart matrix can be identified by the derivative of SE PDF.

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.

This article provides necessary and sufficient conditions that characterize independence among sub-vectors of a singular multivariate skew-normal random vector.

Random matrix theory (RMT) is a sophisticated technique to analyze the cross-correlations of multivariate time series, while it suffers from the limitation on characterizing the linear relationships. In this paper, we propose a new mutual-information matrix analysis to study the nonlinear interactions of multivariate time series, including: (i) The N-dimensional mutual information ranging between 0 and 1 can describe the strength of nonlinear interactions. (ii) The eigenvalues of the random mutual-information matrix yield the Marchenko–Pastur distribution, except that the dominant eigenvalue is significantly larger than the other eigenvalues. (iii) The distribution of most eigenvectors components of the random mutual-information matrix subjects to the Gaussian distribution, while the dominant eigenvector components tend to follow the uniform distribution. A large value of the N-dimensional mutual information, and the deviations from the eigenvalues distribution as well as the distribution of the eigenvectors components both imply the presence of interactions among the underlying time series. In the empirical analysis, we design a simulation which reveals the advantages of the mutual-information analysis over the RMT. We also apply the mutual-information matrix analysis to a real-world application that indicates the presence of interactions among the stock time series.

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 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.

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.

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.

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.

The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.

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.

The paper is devoted to an extension of the multivariate Matsumoto-Yor (MY) independence property with respect to a tree with p vertices to the case where random variables corresponding to the vertices of the tree are replaced by random matrices. The converse of the p -variate MY property, which characterizes the product of one gamma and p - 1 generalized inverse Gaussian distributions, is extended to characterize the product of the Wishart and p - 1 matrix generalized inverse Gaussian distributions.

This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore–Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik type class of shrinkage estimators. Based on the unbiased risk estimator, a sufficient condition for the minimaxity is expressed not only as a differential inequality, but also as an integral inequality. Also, generalized Bayes minimax estimators are established by using an interesting structure of singular multivariate normal distribution.

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.

We consider a multivariate linear response regression in which the number of responses and predictors is large and comparable with the number of observations, and the rank of the matrix of regression coefficients is assumed to be small. We study the distribution of singular values for the matrix of regression coefficients and for the matrix of predicted responses. For both matrices, it is found that the limit distribution of the largest singular value is a rescaling of the Tracy-Widom distribution. Based on this result, we suggest algorithms for the model rank selection and compare them with the algorithm suggested by Bunea, She and Wegkamp. Next, we design two consistent estimators for the singular values of the coefficient matrix, compare them, and derive the asymptotic distribution for one of these estimators.., 31 page

In this article, main characteristics, marginal, joint, and conditional inferences of a generalized multivariate Gumbel model are derived, and random vector generation is described. Distribution of the sum where summands come from a bivariate generalized multivariate Gumbel distribution is derived.

In this paper, we improve upon some observations made in recent papers on the subject of increasingly flat interpolation. We shall establish that the corresponding Lagrange functions converge both for a finite set of functions (collocation matrix) and also for kernels (Fredholm matrix). In our analysis, we use a finite Maclaurin expansion of a multivariate function with remainder and some additional matrix theoretic facts.

We consider computational methods for evaluating and approximating multivariate chi-square probabilities in cases where the pertaining correlation matrix or blocks thereof have a low-factorial representation. To this end, techniques from matrix factorization and probability theory are applied. We outline a variety of statistical applications of multivariate chi-square distributions and provide a system of MATLAB programs implementing the proposed algorithms. Computer simulations demonstrate the accuracy and the computational efficiency of our methods in comparison with Monte Carlo approximations, and a real data example from statistical genetics illustrates their usage in practice.

The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution function (CDF) of the ‘smallest’ eigenvalue is obtained, but also the CDF of the ‘largest’ eigenvalue and the probability density functions of both are expressed in a similar and convenient matrix form. These compact expressions involve only inner products of exponential vectors, vectors of monomials and certain coefficient matrices which therefore assume a key role of carrying all the required information to build the expressions. The computation of these all-important coefficient matrices involves the evaluation of a determinant of a Hankel matrix of incomplete gamma functions. They offer a theorem which proves that the latter matrix has ‘catalectic’ properties, such that the degree of its determinant is surprisingly small. The theorem also implies a closed-form and numerical procedure (no symbolic calculations required) to build the coefficient matrices.

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.

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.

This paper focuses on Bezout equations derived from multivariate polynomial matrices in which relationships between one primary variable and other variables are described by real entire functions. We propose a method for obtaining a solution belonging to a set of multivariate rational function matrices in which all entries are real entire functions with respect to the primary variable. The proposed method is based on a new approach that overcomes the constraints and difficulties due to many variables by expanding a class of solutions to multivariate rational function matrices.

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 trace of a Wishart matrix, either central or non-central, has important roles in various multi-variate statistical questions. We review several expressions of its distribution given in the literature, establish some new results and provide a discussion on computing methods on the distribution of the ratio: the largest eigenvalue to trace.

The skewness of a matrix quadratic form XX 0 is obtained using the expectation of stochastic matrix and applying the properties of commutation matrices, where X Np;n (0; ;In).

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.

This paper introduces matrix-variate $t$-distributions for which degree of freedom is a multivariate parameter. A relation for a density function is obtained.

This article provides two copula families on [0, 1]n obtained from the Laplace transforms of the multivariate gamma distribution and the multifactor gamma distribution given by and , respectively, where P is an affine polynomial with respect to the n variables θ1, …, θn.These copulas allow in particular to obtain multivariate gamma distributions for which the cumulative distribution functions and the probability distribution functions are known.

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.

This note points out some flaws in an article by Chakraborty and Chatterjee (Sankhya: The Indian Journal of Statistics, 75-B, 1, 1–15, 2013) in which formulae for the mean vector, dispersion matrix and the moment generating function of multivariate folded normal distribution are derived. It is observed that the formulae and their derivation are incorrect. Correct formula of the mean for general case, and formula for the dispersion for the special case where the mean of the parent distribution is zero are stated.

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.