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Developments in X-ray scattering instruments have led to unprecedented access toin situand parametric X-ray scattering data. Deriving scientific insights and understanding from these large volumes of data has become a rate-limiting step. While formerly a data-limited technique, pair distribution function (PDF) measurement capacity has expanded to the point that the method is rarely limited by access to quantitative data or material characteristics – analysis and interpretation of the data can be a more severe impediment. This paper shows that multivariate analyses offer a broadly applicable and efficient approach to help analyse series of PDF data from high-throughput andin situexperiments. Specifically, principal component analysis is used to separate features from atom–atom pairs that are correlated – changing concentration and/or distance in concert – allowing evaluation of how they vary with material composition, reaction state or environmental variable. Without requiring prior knowledge of the material structure, this can allow the PDF from constituents of a material to be isolated and its structure more readily identified and modelled; it allows one to evaluate reactions or transitions to quantify variations in species concentration and identify intermediate species; and it allows one to identify the length scale and mechanism relevant to structural transformations.

We present in this note a sufficient condition for sums of TP 2 functions (totally positive of order 2) to be TP 2 . We apply our results to several distributions and provide, in particular, conditions for which certain widely used families of distributions are multivariate totally positive of order 2.

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

Generalized exponential distribution has received some attention in the last few years. Recently, Kundu and Gupta (Advances in Statistical Analysis, 95, 169–185, 2011) and Shoaee and Khorram (Journal of Statistical Planning and Inference, 142, 2203–2220, 2012) introduced an absolute continuous bivariate generalized exponential distribution. In this paper, we propose an absolute coinuous multivariate generalized exponential distribution. The proposed distribution is very flexible, and the joint probability density functions can take different shapes. We provide several properties of this model. Further, it is observed that the multivariate generalized exponential model can be obtained using multivariate Clayton copula. The maximum likelihood estimators are quite difficult to compute in practice. Due to this reason, we propose two step estimation procedure using the copula approach, which are quite easy to implement. Simulation experiments are performed to compare the performances of the two different estimators, and the performances are quite similar in nature particularly for large sample sizes. One multivariate bone mineral density data set has been analyzed for illustrative purposes, and it is observed that the proposed model provides a very good fit to the data set.

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.

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

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

Mixtures of normal distributions provide a useful modeling extension of the normal distribution—both univariate and multivariate. Unlike the normal distribution, mixtures of normals can capture the kurtosis (fat tails) and nonzero skewness often necessary for accurately modeling a variety of real-world variables. An efficient analytical Monte Carlo method is proposed for considering multivariate mixtures of normal distributions having arbitrary covariance matrices. The method consists of a linear transformation of a multivariate normal having a computed covariance matrix into the desired multivariate mixture of normal distributions. The computed covariance matrix is derived analytically. Among the properties of the multivariate mixture of normals that we demonstrate is that any linear combination of mixtures of normal distributions is also a mixture of normal distributions. Methods of fitting mixtures of normal distributions are briefly discussed. A motivating example carried throughout this paper is the use of multivariate mixtures of normals for modeling daily changes in market variables.

Motivated from the stochastic representation of the univariate zero-inflated Poisson (ZIP) random variable, the authors propose a multivariate ZIP distribution, called as Type I multivariate ZIP distribution, to model correlated multivariate count data with extra zeros. The distributional theory and associated properties are developed. Maximum likelihood estimates for parameters of interest are obtained by Fisher's scoring algorithm and the expectation-maximization (EM) algorithm, respectively. Asymptotic and bootstrap confidence intervals of parameters are provided. Likelihood ratio test and score test are derived and are compared via simulation studies. Bayesian methods are also presented if prior information on parameters is available. Two real data sets are used to illustrate the proposed methods. Under both AIC and BIC, our analysis of the two data sets supports the Type I multivariate zero-inflated Poisson model as a much less complex alternative with feasibility to the existing multivariate ZIP models proposed by Li et?al. (Technometrics, 29-38, Vol 41, 1999).

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.

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

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

We give the cumulative distribution function (cdf) of M n , the (element-wise) maximum of a sequence of n observations from a multivariate AR ( p ) process. We do the same for a multivariate MA ( p ) process. Solutions are first given in terms of repeated integrals and then for the case, where the marginal cdf of the observations is absolutely continuous. The cdf of the multivariate maximum M n is then given as a weighted sum of the n th powers of the eigenvalues of a non-symmetric Fredholm kernel. The weights are given in terms of the left and right eigenfunctions of the kernel.

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.

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

The relationship between two sets of real variables defined for the same individuals can be evaluated by few different correlation coefficients. For the functional data we have only one important tool: the canonical correlations. It is not immediately straightforward to extend other similar measures to the context of functional data analysis. In this work we show how to use commonly known measures of correlation for two sets of variables: \(\mathop{\mathrm{rV}}\nolimits\) coefficient and distance correlation coefficient for multivariate functional case. Finally, these three different coefficients are compared and their use is demonstrated on two real examples.

In this chapter, our objective is twofold. First, we establish a connection between the stable distributions with fractional calculus. This is accomplished by defining appropriate fractional diffusion equations, the fundamental solution of which provides the PDF for the univariate and multivariate stable distributions.1 Second, by using some analytic-numerical approaches such as the homotopy perturbation method, the Adomian decomposition method, and the variational iteration method, which are used for solving partial differential equations (PDEs), we obtain some analytic-numerical approximations for the PDF of the univariate and multivariate stable distributions.

We propose estimators for the tail index and the spectral measure of multivariate $$\alpha $$ -stable distributions and derive their asymptotic properties. Simulation studies reveal the appropriateness of the estimators. Applications to financial data are also considered.

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.

Several fascinating examples of non-Gaussian bivariate distributions which have marginal distribution functions to be Gaussian have been proposed in the literature. These examples often clarify several properties associated with the normal distribution. In this paper, we generalize this result in the sense that we construct a p -dimensional distribution for which any proper subset of its components has the Gaussian distribution. However, the joint p -dimensional distribution is inconsistent with the distribution of these subsets because it is not Gaussian. We study the probabilistic properties of this non-Gaussian multivariate distribution in detail. Interestingly, several popular tests of multivariate normality fail to identify this p -dimensional distribution as non-Gaussian. We further extend our construction to a class of elliptically contoured distributions as well as skewed distributions arising from selections, for instance the multivariate skew-normal distribution.

In this 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 have derived the distribution of multivariate order statistics for multivariate exponential & multivariate weibull distribution. The moment expression for multivariate order statistics has also been derived.

In this paper, we show that the derivation of Lemma 3 of Das and Dey (2010) needs to be corrected by using a logical transformation, instead of the ad-hoc transformation which is partially motivated by its univariate equivalent transformation. The correct derivation is presented by two approaches.

In this article, we introduce a new three-parameter skewed distribution of which normal distribution is a special case. This distribution is obtained by using geometric sum of independent identically distributed normal random variables. We call this distribution as the geometric skew normal distribution. Different properties of this new distribution have been investigated. The probability density function of geometric skew normal distribution can be unimodal or multimodal, and it always has an increasing hazard rate function. It is an infinite divisible distribution, and it can have heavier tails. The maximum likelihood estimators cannot be obtained in explicit forms. The EM algorithm has been proposed to compute the maximum likelihood estimators of the unknown parameters. One data analysis has been performed for illustrative purposes. We further consider multivariate geometric skew normal distribution and explore its different properties. The proposed multivariate model induces a multivariate Levy process, and some properties of this multivariate process have been investigated. Finally, we conclude the paper.

In this paper we first introduce a new class of multivariate generalized asymmetric skew-normal distributions with two parameters λ1,λ2 that we present it by BGSNn, m (λ1,λ2), and we finally obtain some special properties of BGSNnm(λ1,λ2).

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.

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

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.

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

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

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.

Like some other multivariate skew normal distributions, the unified skew normal (SUN) distribution preserves important properties of the normal distribution. In this article, we show that for a random vector with a unified multivariate skew normal distribution all column (row) full rank linear transformations are in the same family of distributions. Using this property, we provide a characterization for the SUN distribution. In addition, we show that the joint distribution of the independent SUN random vectors is again a SUN distributed random vector. With this property and closure under the linear transformation, we finally show the closure of sums of independent SUN random vectors, and as expected it belongs to the same family.

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.

More than one series of longitudinal data frequently encountered in biomedical, psychological and clinical research are routinely analyzed under a multivariate linear mixed model framework with underlying multivariate normality assumptions for the random effects and within-subject errors. However, such normality assumption might not offer robust inference if the data, even after being transformed, particularly exhibit skewness. In this paper, we propose a multivariate skew-normal linear mixed model constructed by assuming a multivariate skew-normal distribution for the random effects and a multivariate normal distribution for the random errors. A damped exponential correlation structure is adopted to address the within-subject autocorrelation possibly existing among irregularly observed measures. We present an efficient alternating expectation-conditional maximization (AECM) algorithm for maximum likelihood estimation of parameters. The techniques for estimation of random effects and prediction of future outcomes are discussed. Our proposed model is motivated by, and used for, the analysis of AIDS clinical trials in which we investigate the ‘association-of-the-evolutions’ and the ‘evolution-of-the-association’ of HIV-1 RNA copies and CD4+T cell counts during antiviral therapies.

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

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

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

This paper presents a robust probabilistic mixture model based on the multivariate skew-t-normal distribution, a skew extension of the multivariate Student's t distribution with more powerful abilities in modelling data whose distribution seriously deviates from normality. The proposed model includes mixtures of normal, t and skew-normal distributions as special cases and provides a flexible alternative to recently proposed skew t mixtures. We develop two analytically tractable EM-type algorithms for computing maximum likelihood estimates of model parameters in which the skewness parameters and degrees of freedom are asymptotically uncorrelated. Standard errors for the parameter estimates can be obtained via a general information-based method. We also present a procedure of merging mixture components to automatically identify the number of clusters by fitting piecewise linear regression to the rescaled entropy plot. The effectiveness and performance of the proposed methodology are illustrated by two real-life examples.