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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...

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.

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.

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.

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.

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 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.

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.

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.

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...

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, 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 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.

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 study the problem of estimating the covariance matrix of stationary multivariate time series based on the minimum density power divergence method that uses a multivariate skew $$t$$ distribution family. It is shown that under regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. A simulation study is provided for illustration.

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.

When we consider a statistical test in the high dimensional case, we often need estimators of the functions of the covariance matrix @S. Especially, it is needed to estimate a"2=(1/p)[email protected]^2. The unbiased and consistent estimator of a"2 is proposed in preceding study when the population distribution is multivariate normal. But it is difficult to estimate in the non-normal case. So we propose the unbiased and consistent estimators for some functions of covariance matrix including a"2 under the non-normal case. Through the numerical simulation, we confirmed the accuracy of the approximation of our proposed estimators. Using proposed estimators, we proposed a test for assessing multivariate normality of the high-dimensional data.

We consider regularization of the parameters in multivariate linear regression models with the errors having a multivariate skew-t distribution. An iterative penalized likelihood procedure is proposed for constructing sparse estimators of both the regression coefficient and inverse scale matrices simultaneously. The sparsity is introduced through penalizing the negative log-likelihood by adding L1-penalties on the entries of the two matrices. Taking advantage of the hierarchical representation of skew-t distributions, and using the expectation conditional maximization (ECM) algorithm, we reduce the problem to penalized normal likelihood and develop a procedure to minimize the ensuing objective function. Using a simulation study the performance of the method is assessed, and the methodology is illustrated using a real data set with a 24-dimensional response vector. & 2014 Elsevier B.V. All rights reserved.

A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.

This paper derives the risk functions of a class of shrinkage estimators for the mean parameter matrix of a matrix variate elliptically contoured distribution. It is showed that the positive rule shrinkage estimator outperformed the shrinkage and unrestricted (maximum likelihood) estimators. To illustrate the findings of the paper, the relative risk functions for different degrees of freedoms are given for a multivariate t distribution. Shrinkage estimators for the matrix variate regression model under matrix normal, matrix t or Pearson VII error distributions would be special cases of this paper.

Multivariate partially linear models are generalizations of univariate partially linear models. In the literature, some estimators of treatment effects and nonparametric components have been proposed. In this note, the estimator of the covariance matrix in multivariate partially linear models is derived and some of its properties are given.

In this article we study the asymptotic behavior of the distributions of multivariate generalized Cox processes with nonrandom centering and some special scalar normalization to a mixture of multivariate normal distribution as t → ∞. In this special case necessary and sufficient conditions for convergence to a limit distribution had been found.

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.

In this paper, we propose a multivariate log-linear Birnbaum–Saunders regression model. We discuss maximum-likelihood estimation of the model parameters and provide closed-form expressions for the score function and for Fisher's information matrix. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum-likelihood estimator. Some influence methods, such as the local influence and generalized leverage are discussed and the normal curvatures for studying local influence are derived under some perturbation schemes. Further, a test for the homogeneity of the shape parameter of the multivariate regression model is investigated. A real data set is presented for illustrative purposes.

In the case of informative sampling the sampling scheme explicitly or implicitly depends on the response variables. As a result, neither the sample distribution of response variables, nor the covariance matrix reects the corresponding population counterparts. In this paper, a relationship between multivariate sample and population distributions is used. Based on this, the influence of the informative sampling on the covariance matrix is investigated. It is shown that with inclusion probabilities in a multiplicative form with respect to study variables, the independence between variables is preserved in the sample. Further, it is shown that with inclusion probabilities exponentially depending on the study variables, the multivariate exponential family is invariant under sampling. The sample distribution belongs to the same family as the population distribution but with different parameters. The relationship between parameters is given. The multinomial and multivariate normal distributions are examined in more detail and the parameters of their sample distributions are derived explicitly. The effect of the informative sampling on the respective covariance matrices and correlations is analysed and illustrated in the examples.

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.

In this study, we discuss two kinds of multivariate two-sided tests for normal mean vectors with unknown covariance matrix. First, assuming that all components of a normal mean vector are simultaneously nonnegative or non positive, we consider a multivariate two-sided test for testing whether the normal mean vector is equal to zero or not. Next, assuming that all differences of components between two normal mean vectors are simultaneously non negative or non positive, we consider a multivariate two-sided test for testing whether the two normal mean vectors are equal or not. We construct methods for testing by referring to Glimm et al. (2002), Tamhane and Logan (2002) and Sasabuchi (2007). Finally, we give some simulation results regarding critical values and power of the test intended to compare the three methods.

In this paper, we study the robust estimation for the covariance matrix of stationary multivariate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) designed by Basu et?al. (1998). To supplement the result of Kim and Lee (2011), we employ a multivariate normal mixture family instead of a multivariate normal family. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration. A real data analysis applied to the portfolio selection problem is also considered.

In this article, we study several properties of the inverted matrix variate gamma distribution. Further, Bayes estimators using conjugate prior knowledge under square error loss function are also derived.

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.

Baysian inference is considered for the precision matrix of the multivariate regression model with distribution of the random responses belonging to the multivariate scale mixtures of normal distributions. The posterior distribution and some identities involving expectations taken with respect to this posterior distribution are derived when the prior distribution of the parameters is from the conjugate family. The results are specialized to the case where the random responses have a matrix-t distribution and thus generalizing the results of Zellner (1976) and Muirhead (1986).

The need to simulate from a positive multivariate normal distribution arises in several settings, specifically in Bayesian analysis. A variety of algorithms can be used to sample from this distribution, but most of these algorithms involve Gibbs sampling. Since the sample is generated from a Markov chain, the user has to account for the fact that sequential draws in the sample depend on one another and that the sample generated only follows a positive multivariate normal distribution asymptotically. The user would not have to account for such issues if the sample generated was i.i.d. In this paper, an accept-reject algorithm is introduced in which variates from a positive multivariate normal distribution are proposed from a multivariate skew-normal distribution. This new algorithm generates an i.i.d. sample and is shown, under certain conditions, to be very efficient.

In this article, we consider the problem of testing a general multivariate linear hypothesis in a multivariate linear model when the N × p observation matrix is normally distributed with unknown covariance matrix, and N ≤ p. This includes the case of testing the equality of several mean vectors. A test is proposed which is a generalized version of the two-sample test proposed by Srivastava and Du (2008). The asymptotic null and nonnull distributions are obtained. The performance of this test is compared, theoretically as well as numerically, with the corresponding generalized version of the two-sample Dempster (1958) test, or more appropriately Bai and Saranadasa (1996) test who gave its asymptotic version.

In this paper, it is shown that a complex multivariate random variable is a complex multivariate normal random variable of dimensionality if and only if all nondegenerate complex linear combinations of have a complex univariate normal distribution. The characteristic function of has been derived, and simpler forms of some theorems have been given using this characterization theorem without assuming that the variance-covariance matrix of the vector is Hermitian positive definite. Marginal distributions of have been given. In addition, a complex multivariate t-distribution has been defined and the density derived. A characterization of the complex multivariate t-distribution is given. A few possible uses of this distribution have been suggested.