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

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

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

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...

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.

Multivariate process capability indices are applied to account the capability of the processes which the quality of the products depends on two or more related characteristics. We call a tolerance region asymmetric, when the target value of at least one characteristic is not the mid-point of the tolerance interval. This paper introduces a superstructure index to measure the capability of multivariate normal process in asymmetric tolerance regions, which could be applied for symmetric cases, too. In addition, the effects of two modification factors in the index which weigh the mean departure from target and process variability are investigated. Furthermore, some examples are presented to demonstrate the applicability and effectiveness of the proposed index.

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.

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...

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.

A multivariate normal mean–variance mixture based on a Birnbaum–Saunders (NMVMBS) distribution is introduced and several properties of this new distribution are discussed. A new robust non-Gaussian ARCH-type model is proposed in which there exists a relation between the variance of the observations, and the marginal distributions are NMVMBS. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of this normal mean–variance mixture distribution is given. A simulation study and some real data are used to demonstrate the modelling strength of this new model.

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.

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the...

In this article, an estimation problem for multivariate stable laws using wavelets has been studied. The method of applying wavelets, which has already been done, to estimate parameters in univariate stable laws, has been extended to multivariate stable laws. The proposed estimating method is based on a nonlinear regression model on wavelet coefficients of characteristic functions. In particular, two parametric sub-classes of stable laws are considered: the class of multivariate stable laws with discrete spectral measure, and sub-Gaussian laws. Using a simulation study, the proposed method has been compared with well-known estimation procedures.

Several measures of process yield, defined on univariate and multivariate normal process characteristics, have been introduced and studied by several authors. These measures supplement several well-known Process Capacity Indices (PCI) used widely in assessing the quality of products before being released into the marketplace. In this paper, we generalise these yield indices to the location-scale family of distributions which includes the normal distribution as one of its member. One of the key contributions of this paper is to demonstrate that under appropriate conditions, these indices converge in distribution to a normal distribution. Several numerical examples will be used to illustrate our procedures and show how they can be applied to perform statistical inferences on process capability.

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.

The univariate fatigue life distribution proposed by Birnbaum and Saunders [A new family of life distributions. J Appl Probab. 1969;6:319–327] has been used quite effectively to model times to failure for materials subject to fatigue and for modelling lifetime data and reliability problems. In this article, we introduce a Birnbaum–Saunders (BS) distribution in the multivariate setting. The new multivariate model arises in the context of conditionally specified distributions. The proposed multivariate model is an absolutely continuous distribution whose marginals are univariate BS distributions. General properties of the multivariate BS distribution are derived and the estimation of the unknown parameters by maximum likelihood is discussed. Further, the Fisher's information matrix is determined. Applications to real data of the proposed multivariate distribution are provided for illustrative purposes.

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

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.

Based on the multivariate spatial rank function introduced by Mottonen and Oja [(1995), ‘Multivariate Spatial Sign and Rank Methods’, Journal of Nonparametric Statistics, 5, 201–213] and Mottonen et al. [(1997), ‘On the Efficiency of Multivariate Spatial Sign and Rank Tests’, Annals of Statistics, 25, 542–552], an extension of the univariate Wilcoxon regression estimate to multivariate linear models is proposed and studied. For both of the cases covariates are deterministic and i.i.d. random: we show that the proposed estimate is consistent and asymptotically normal under some appropriate assumptions. We have demonstrated that the asymptotic relative efficiency of the new regression estimate is the same as that of the generalised multivariate Hodges–Lehmann location estimates proposed by Chaudhuri [(1992), ‘Multivariate Location Estimation Using Extension of R-estimates Through U-statistics Type Approach’, Annals of Statistics, 20, 897–916] (with m=2); thus it possesses high efficiency. Simulations show t...

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.

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time. In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.

This article introduces a class of weighted multivariate t-distributions, which includes the multivariate generalized Student t and multivariate skew t as its special members. This class is defined as the marginal distribution of a doubly truncated multivariate generalized Student t-distribution and studied from several aspects such as weighting of probability density functions, inequality constrained multivariate Student t-distributions, scale mixtures of multivariate normal and probabilistic representations. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.

In this paper, form-invariant weighted distributions are considered in an exponential family. The class of bivariate distribution with invariant property is identified under exponential weight function. The class includes some of the custom bivariate models. The form-invariant multivariate normal distributions are obtained under a quadratic weight function.

In this article we define a class of multivariate tolerance regions that turn out to be more resistant than the classical ones to outliers. The tolerance factors are numerically evaluated under the central model, and the sensitivity to deviations from the normal distribution for moderate samples is studied through a Monte Carlo study. Moreover, the influence function of the coverage probability allows us to compare the sensitivity of different proposals to anomalous data. Finally, real data examples are discussed.

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.

This paper considers Statistical Process Control (SPC) when the process measurement is multivariate. In the literature, most existing multivariate SPC procedures assume that the in-control distribution of the multivariate process measurement is known and it is a Gaussian distribution. In applications, however, the measurement distribution is usually unknown and it needs to be estimated from data. Furthermore, multivariate measurements often do not follow a Gaussian distribution (e.g., cases when some measurement components are discrete). We demonstrate that results from conventional multivariate SPC procedures are usually unreliable when the data are non-Gaussian. Existing statistical tools for describing multivariate non-Gaussian data, or transforming the multivariate non-Gaussian data to multivariate Gaussian data, are limited, making appropriate multivariate SPC difficult in such cases. In this paper, we suggest a methodology for estimating the in-control multivariate measurement distribution when a se...

For the data from multivariate t distributions, it is very hard to make an influence analysis based on the probability density function since its expression is intractable. In this paper, we present a technique for influence analysis based on the mixture distribution and EM algorithm. In fact, the multivariate t distribution can be considered as a particular Gaussian mixture by introducing the weights from the Gamma distribution. We treat the weights as the missing data and develop the influence analysis for the data from multivariate t distributions based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Several case-deletion measures are proposed for detecting influential observations from multivariate t distributions. Two numerical examples are given to illustrate our methodology.

It is well known that many data, such as the financial or demographic data, exhibit asymmetric distributions. In recent years, researchers have concentrated their efforts to model this asymmetry. Skew normal model is one of such models that are skew and yet possess many properties of the normal model. In this paper, a new multivariate skew model is proposed, along with its statistical properties. It includes the multivariate normal distribution and multivariate skew normal distribution as special cases. The quadratic form of this random vector follows a χ2 distribution. The roles of the parameters in the model are investigated using contour plots of bivariate densities.

We give a general procedure to characterize multivariate distributions by using products of the hazard gradient and mean residual life components. This procedure is applied to characterize multivariate distributions as Gumbel exponential, Lomax, Burr, Pareto and generalized Pareto multivariate distributions. Our results extend the results of several authors and can be used to study how to extend univariate models to the multivariate set-up.

SYNOPTIC ABSTRACTIn this paper, we proffer a new multivariate gamma distribution with potential applications in survival and reliability modeling. The multivariate distribution is not necessarily restricted to those with gamma marginal distributions. We provide and characterize a generalized location scale family of multivariate gamma distributions. This family possesses three-parameter gamma marginals (in most cases) and it contains absolutely continuous classes, as well as, the Marshall Olkin type of distributions with a positive probability mass on a set of measure zero. Maximum likelihood estimators are developed in the bivariate case.

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 article we consider a generalization of the univariate g-and-h distribution to the multivariate situation with the aim of providing a flexible family of multivariate distributions that incorporate skewness and kurtosis. The approach is to modify the underlying random variables and their quantiles, directly giving rise to a family of distributions in which the quantiles rather than the densities are the foci of attention. Using the ideas of multivariate quantiles, we show how to fit multivariate data to our multivariate g-and-h distribution. This provides a more flexible family than the skew-normal and skew-elliptical distributions when quantiles are of principal interest. Unlike those families, the distribution of quadratic forms from the multivariate g-and-h distribution depends on the underlying skewness. We illustrate our methods on Australian athletes data, as well as on some wind speed data from the northwest Pacific.

We study some properties for multivariate weighted distributions related to reliability measures, ordering, characterization and dependence properties, by compiling and extending previous results given by different authors. We pay special attention to the multivariate size biased and equilibrium distributions, and propose a new definition for the multivariate equilibrium distribution.

Kendall’s τ for univariate samples is extended to the multivariate case using unit direction vectors in place of signs. Depending on whether the design matrix is fixed, this statistic yields a test for the multivariate analysis of variance or multivariate regression. Tests of independence based on this statistic are easy, efficient, and robust. Its asymptotic properties and its Pitman efficiency are studied. In addition, some useful algebra in multiple dimensions is 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

A new affine invariant multivariate analogue of the two-sample Mann–Whitney–Wilcoxon test based on the Oja criterion function is introduced. The associated affine equivariant estimate of shift, the multivariate Hodges-Lehmann estimate, is also considered. Asymptotic theory is developed to provide approximations for null distribution as well as for a sequence of contiguous alternatives to consider limiting efficiencies of the test and estimate. The theory is illustrated by an example. Hettmansperger et al. [9] considered alternative slightly different affine invariant extensions also based on the Oja criterion. The methods proposed in this paper are computationally more intensive, but surprisingly more efficient in the multivariate normal case. For elliptical distributions, the limiting efficiencies coincide with those of the affine invariant spatial rank methods.

In this paper, we define multivariate skew t-distribution which has some of the properties of multivariate t-distribution and has a shape parameter to represent skewness. Some of its properties are also studied including the moments. Multivariate skew-Cauchy distribution is given as a special case.

The Kotz-type distribution was introduced by Kotz (1975) as a generalization of the multivariate normal distribution. Since 1990 there has been a surge of activity relating to this distribution. We have identified some 25 papers on the Kotz-type distribution over the period from 1990 to 2002 - compared to just 5 over the period from 1980 to 1989. The aim of this paper is to review the developments in the following areas: marginal distributions; moments; characteristic functions; characterizations; asymptotics; quadratic forms; estimation; hypothesis testing; generalizations; Bayesian inference; and, applications in other areas such as ecology, discriminant analysis, mathematical finance, repeated measurements, shape theory and signal processing. We feel that this review could be important as a source of reference and for unlocking further research on the distribution.

We propose a test of dimension in multivariate regression. This test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model, but it does not require a prespecified model. The test may be particularly useful at the outset of an analysis before a multivariate model is posited, because it can lead to low-dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function.