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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...

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.

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.

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.

In this article, we introduce a new form of distribution whose components have the Poisson or Skellam marginal distributions. This new specification allows the incorporation of relevant information on the nature of the correlations between every component. In addition, we present some properties of this distribution. Unlike the multivariate Poisson distribution, it can handle variables with positive and negative correlations. It should be noted that we are only interested in modeling covariances of order 2, which means between all pairs of variables. Some simulations are presented to illustrate the estimation methods. Finally, an application of soccer teams data will highlight the relationship between number of points per season and the goal differential by some covariates.

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization the well-known goodness of link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (XT1, X2T, …, XTk, ZT)T = (X11, …, X1n, …, Xk1, …, Xkn, Z1, …, Zn)T and derive the distribution of concomitant of multivariate order statistics arising from X1, X2, …, Xk. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.

The class of stable distributions plays a central role in the study of asymptotic behavior of normalized partial sums, the same role performed by normal distribution among those with finite second moment. In this note, by exploiting the connection between stable laws and regularly varying functions, we present weighted similarity tests for stable location-scale families. The proposed weight functions are based on the 2nd-order Mallows distance between the empirical distribution and the root stable distribution. And the resulting statistics converge weakly to functionals of Brownian bridge.

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.

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.

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 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.

This note mainly aims to illustrate that some quadratic problems are robust in a sense with respect to the probabilistic distributions involved. The secondary moments of the quadratic forms of a multivariate t distribution are calculated. Then, the resulting formulae are applied to the quadratic problems of quadratic sufficiency and quadratic prediction. It is shown by revisiting the two problems that the same conclusions hold when the multivariate normal distribution is replaced with a multivariate t distribution.

A general multivariate Weibull distribution is introduced in this article. The GMW is constructed by a well-defined exponent Radon measure which is satisfied by a functional equation with some assumptions. There is rarely literature published regarding the study of characterization of the multivariate Weibull distribution. Yeh (2009) investigated two characterizations on homogeneous multivariate semi-Weibull distribution. Many characterization theorems of the GMW are proved in this article. All these characterizations lead to a multivariate functional equation. Finally, the limiting distribution of the normalized geometric minima of the GMW is discerned as the general multivariate Pareto distribution.

This article proposes a class of multivariate bilateral selection t distributions useful for analyzing non-normal (skewed and/or bimodal) multivariate data. The class is associated with a bilateral selection mechanism, and it is obtained from a marginal distribution of the centrally truncated multivariate t. It is flexible enough to include the multivariate t and multivariate skew-t distributions and mathematically tractable enough to account for central truncation of a hidden t variable. The class, closed under linear transformation, marginal, and conditional operations, is studied from several aspects such as shape of the probability density function, conditioning of a distribution, scale mixtures of multivariate normal, and a probabilistic representation. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.

Consider an estimation problem of a linear combination of population means in a multivariate normal distribution under LINEX loss function. Necessary and sufficient conditions for linear estimators to be admissible are given. Further, it is shown that the result is an extension of the quadratic loss case as well as the univariate normal case.

This article considers the nonparametric maximum likelihood estimator (NPMLE) of a joint distribution function when the multivariate failure times of interest are interval-censored. With different types of interval censoring mechanism, the NPMLE's of the multivariate distribution function are studied and the strong consistency for the NPMLEs is obtained in terms of a self-consistency equation. Furthermore, the convergence rate of the estimator is given, which depends on the types of interval censoring mechanism.

The multivariate split normal distribution extends the usual multivariate normal distribution by a set of parameters which allows for skewness in the form of contraction/dilation along a subset of the principal axes. This article derives some properties for this distribution, including its moment generating function, multivariate skewness, and kurtosis, and discusses its role as a population model for asymmetric principal components analysis. Maximum likelihood estimators and a complete Bayesian analysis, including inference on the number of skewed dimensions and their directions, are presented.

In this paper, we study two time-dependent association measures for the bivariate as well as the multivariate normal distribution. In the case of bivariate normal distribution, we obtain a necessary and sufficient condition for the distribution to be right corner set increasing (RCSI). For the multivariate normal distribution, a sufficient condition, in terms of partial regression coefficients, for the distribution to be RCSI is obtained.The actual values of these measures are derived for the bivariate as well as the multivariate normal distribution. Some of the expressions are complex but are useful in reliability studies

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Ot...

The Hotelling's T2statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T2statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T2. In this paper, we use a kernel smoothing technique to estimate the distribution of the T2statistic as well as of the UCL of the T2chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T2control chart for sample observations taken f...

Bayesian multivariate procedure is introduced to assess the intraclass correlation coefficients under the multivariate mixed linear model. Unified estimators for the multivariate measures are proposed as the conditional posterior means of multivariate inverted matrix Dirichlet distribution. These estimators have explicit expressions and generalize the results of case of variance components. A real data example is also provided.

The heceroscedastic multivariate linear model with multivariate normal error distribution has been considered, using the structural relation of the model, the prediction distribution of future responses of the model has been derived. it is observed that for known covariance parameters the prediction distribution of the model has a product of m multivariate Student t distribution. It is to be noted that the prediction distribution for the Student t error also has a product of m multivariate Student t distribution. Some special cases have been discussed.

A multivariate generation of Bose's ( 1935) bivariate chi-square distribution is presented. This multivariate family arises for some problems involving the comparison of k>I experimental treatments to a control when the limiting distribution of the twosample statistic is chi-square. 1nference for these comparison with control problems is based on the null distribut~on dfthe maximum component. A computation algorithm and tables for probability points are presented for the distribution of the maximum component. Two applications are discussed.

A multivariate generalized Polya - Eggenberger model has been obtained by computing the probability of a first passage event with the help of combinatorial method involving counting of multi-dimensional lattice paths. The model generates a number of important discrete multivariate distributions both as particular cases and as limiting cases. A recurrence relation among the moments of the model has been established and hence first two moments have been obtained.

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.

In this paper, we introduce a new multivariate pareto (MVP) distribution with many interesting properties. we extend the results of characterization of univariate and bivariate pareto distributions given by Krishnaji (1970) and veenus and Nair (1994) respectively. We also extend the property of dullness of univariate pareto distribution given by Talwalkar (1980) to the multivariate pareto case. We obtain the maximum likelihood estimate (MLE) of the parameters and their asymptotic multivariate normal (AMVN) distrioutions. We propose large sample studentized test for testing independence and identical marginals of the components.

A random vector has a multivariate Pareto distribution if one of its univariate conditional distribution is Pareto and some of its marginals are identically distributed.A general method developed in the course of the proof of this result is applied also to characterize the multivariate Student (Cauchy) measure by one univariate Student conditional distribution.

The concept of weighted distributions is well-known in the literature concerning observational studies and surveys in research related to forestry, ecology, bio-medicine and many other areas (cf. Rao (1965)). This paper extends the idea of weighted distributions to multivariate case. A few multivariate orderings have been defined and some partial ordering results are presented. Some results regarding multivariate positive and negative dependence are also discussed. Multivariate weighted distributions - joint, marginal and conditional have been defined and some important results concerning them are presented along with an illustration. The Multivariate Poisson Negative Hypergeometric Distribution has been derived.

A multivariate generalization, Rv, of von Neumann's ratio is proposed. The null distribution is evaluated under the assumption that observations are independent, identically distributed with a multivariate normal distribution. The Rv statistic provides an intuitively appealing measure of multivariate association which indicates whether the observed serial correlation is positive (the distance between successive observations is less than expected) or negative. The operating characteristics of the corresponding one-sided test is evaluated. A two-sided likelihood ratio test (LR) proposed previously is modified to obtain a one-sided test. Comparisons of these two procedures indicates that both provide accurate control over the size of the test, but that the Rv test appears to be more powerful.

Multivariate Poisson distribution is a well known distribution in multivariate discrete distributions. The author: Kawamura had discussed around the distribution in [1], [2] and shown recurrence relations for the distribution in [3], [4]. . These are the recurrence relations in bivariate case, see [3], [4]. In this paper it will be discussed that a similar relation also holds in the case of generalized multivariate distribution. A relation between algebraic structure of the range space and its probability function concerning the distribution will be investigated in detail, especially,by the recurrencerelations: .

The lower dimensional marginal density functions of a truncated multivariate density function is derived in general, and shown that it is a function of untruncated marginal density function, appropriately defined conditional distribution function and size of the multivariate truncation region. As a special case, lower dimensional marginal density function of a truncated multivariate normal distribution is given.

The exact distributions of the univariate- and multivariate Behrens-Fisher statistics are derived. As is well known, these distributions are dependent on the unknown variances or covariance Matrices.The critical values of the exact distribution are computed and compared to the critical values of several approximations in the univariate case with the variances replaced by sample estimates.In the multivariate situation it is shown that this distribution is a special case of a generalization of a matrix Beta type II distribution as derived by van der Merwe and Nel (1987). Due to the computational intractability of the density function we could not compare it to an approximation.

The distribution(s) of future response(s) given a set of data from an informative experiment is known as prediction distribution. The paper derives the prediction distribution(s) from a linear regression model with a multivari-ate Student-t error distribution using the structural relations of the model. We observe that the prediction distribution(s) are multivariate t-variate(s) with degrees of freedom which do not depend on the degrees of freedom of the error distribution.

Weak convergence results are used to investigate asymptotic properties of Mardia's measure of multivariate kurtosis in the context of assessing multivariate normality.

Ratios of independent central Wishart determinants are useful statistics in multivariate analyses, particularly in the study of multivariate linear models. A method based on the inversion of characteristic functions is outlined for deriving new experessions for the probability distribution functions of the logarithms of these statistics. Accurate tables of the percentiles of these distributions have been obtained covering many bivariate and trivariate cases which have been computed by approximating these expression.

Inequalities for tail probabilities of the multivariate normal distribution are obtained, as a generalization of those given by Feller (1966). Upper and lower bounds are given in the equi-correlated case. For an arbitrary correlation matrix R, an upper bound is obtained, using a result of Slepian (1962) which asserts that certain multivariate normal probabilities are a non-decreasing function of correlations.

A continuous multivariate exponential distribution is introduced which can model a full range of correlation structures and attains the Frechet bounds in the bivariate case, is easy to simulate, arises as a model for reliability and failure due to shocks, and is analogous to the multivariate normal distribution. Two examples are given in which it models data satisfactorily

The papsr considers distributions of collections of ratios of normal variables, The derivation of the joint density is linked to SKI sting literature on absolute, incomplete or truncated moments of multinormals. The distribution function may be expressed as a sum of rectangular multi normal probabilities. When the coefficients of variation of the denominators are close to zero, then a simple transformation of the ratios is approximately inultinormal. An application to Bayesian analysis is included.

Many authors have suggested different methods for constructing meaningful multivariate discrete distributions Kemp and Papageorgiu (1982), Marshall and Olkin (1985). There are some drawbacks to the existing multivariate families because of their restricted covariation patterns Taillie et al. (1979). In this paper is proposed a new statistical model, which will allow more flexibility and will not be constrained by the creation of a bivariate or multivariate version of a specific univariate distribution. The method consists of constructing trivariate models using the following three concepts: the ascending diagonal arrays, the third marginal distribution and the total of variables distribution. The multivariate case will be further derived by a reduction to a number of trivariate models.

In this paper, the joint distribution of (a) a linear and-a quadratic form, and (b) two quadratic forms are obtained when the sample is taken from a mixture of two p-co»ponent multivariate normal distributions with mean JJ. and JJ« respectively and common covarlaece matrix Z # Also the distribution of Hotelling's t2-statistic is obtained when irji- + (1-ir)^ - j) t where w f O^i^l f Is the mixing proportion (contamination, 1-w). Actual values of a (level of significance) when nominal level a Is 0»05 are computed for some particolar combination of parameters* These results show that the si2e of the test can differ greatly even for a contan-ination as small as 0,05 if the Mahalanobis (square) distance Is large and/or the sample size Is large

Classical order statistics are generalized to random samples from continuous multivariate distributions.

The evaluation of multivariate normal probability integrals has led several authors to reductions of certain integrals for special cases of the correlation matrix (ρij) of jointly normal variates. This reduction is obtained by representing the original normal variates as an appropriate linear combination of a larger set of variates. One important special case is ρij=+αiαj (i≠j), where -1≦αi≦1, which as Gupta (1963) noted, has been periodically and independently rediscovered. It is the purpose of this paper (1) to show that an analogous representation holds true for ρij = =−αiαj (i≠j), provided that (2) to generalize the representations for ρij=+αiαj and ρij=−αiαj; and (3) to apply these representations to integrals for equi-coordinate and orthant probabilities. This application provides a method for finding the distribution of the largest of a set of equi-correlated normal variates. The method requires only the evaluation of a simple integral of Grubbs (1950) function which represents the distribution of ...

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.

A question of considerable interest is whether a stable distribution is Gaussian or has infinite variance. Three statistics, the kurtosis, the ratio of one half the range to the sample stan-dard deviation, and a regression-type estimator of the characteristic exponent α are compared in a Monte Carlo power study. In addition, the asymptotic distribution of the third statistic is examined and the powers obtained by using approximate normal theory are compared to the Monte Carlo findings in large samples.

We consider a multivariate Pearson family of distributions. Certain parametric restrictions lead to the multivariate normal distribution. Using this fact we propose a number of asymptotically efficient tests. Through Monte Carlo experiments these tests are compared with some of the existing test procedures. A table is provided from which finite-sample critical points can be obtained.

The growth curve model introduced by potthoff and Roy 1964 is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. The methods currently available for estimating the parameters of this model assume an underlying multivariate normal distribution of errors. In this paper, we discuss tw robst estimators of the growth curve loction and scatter parameters based upon M-estimation techniques and the work done by maronna 1976. The asymptotic distribution of these robust estimators are discussed and a numerical example given.

Asymptotic distributions of regression-type estimators for the parameters of stable distributions am obtained. The asymptotic normalized standard deviations of the estimators are computed for various values of the parameters and various choices of the number of points used in getting the regression estimates.

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.

In this paper we assess the sensitivity of the multivariate extreme deviate test for a single multivariate outlier to non-normality in the form of heavy tails. We find that the empirical significance levels can be markedly affected by even modest departures from multivariate normality. The effects are particularly severe when the sample size is large relative to the dimension. Finally, by way of example we demonstrate that certain graphical techniques may prove useful in identifying the source of rejection for the multivariate extreme deviate test.