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In this chapter we extend the theory from univariate distributions to multivariate formulations for a select few distributions: the Gaussian, lognormal, and gamma distributions. We shall also introduce mixed distribution, which is a combination of Gaussian and lognormal random variables. We shall highlight that not all of the desired properties of the univariate descriptive statistics carry over to the multivariate formulation, but start to lay the foundations for the derivation of many of the data assimilation schemes in later chapters.

HE NORMAL DISTRIBUTION is central to much of statistics. In this chapter and the two following, we develop the normal model from the univariate, bivariate, and then, finally, the more general distribution with an arbitrary number of dimensions.

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag&ndash, Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag&ndash, Leffler distributions are considered in detail. Their relations with multivariate `ordinary&rsquo, Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag&ndash, Leffler and generalized Mittag&ndash, Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value at Risk for multivariate random variables as well as stochastic orders are introduced via the set-valued approach., 30 pages, 11 figures

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, we consider Sarmanov’s multivariate discrete distribution as counting distribution in two multivariate compound models: the first model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov’s distribution can join different types of marginals, we also assume that these marginals belong to Panjer’s class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

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...

Consider the evaluation of model-based functions of cumulative distribution functions that are integrals. When the cumulative distribution function does not have a tractable form but simulation of the multivariate distribution is easily feasible, we can evaluate the integral via a Monte Carlo sample, replacing the model-based distribution function by the empirical distribution function. Given a simulation sample of size N, the naive method uses $$O(N^{2})$$ comparisons to compute the empirical distribution function at all N sample vectors. To obtain faster computational speed when N needs to be large to achieve a desired accuracy, we propose methods modified from the popular merge sort and quicksort algorithms that preserve their average $$O(N\log _{2}N)$$ complexity in the bivariate case. The modified merge sort algorithm can be extended to the computation of a d-dimensional empirical distribution function at the observed values with $$O(N\log _{2}^{d-1}N)$$ complexity. Simulation studies suggest that the proposed algorithms provide substantial time savings when N is large.

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

In some application, quality of product or performance of a process described by some functional relationships between some variables known as multivariate linear profile in the literature. In this paper, we propose Max-MEWMA and Max-MCUSUM control charts for simultaneous monitoring of mean vector and covariance matrix in multivariate multiple linear regression profiles in Phase II. The proposed control charts also have ability to diagnose either the location or variation of the process is responsible for out-of-control signal. The performance of the proposed control charts is compared with existing method through Monte-Carlo simulations. Finally, the applicability of the proposed control charts is illustrated using a real case of calibration application in the automotive industry.

This article introduces a robust extension of the mixture of factor analysis models based on the restricted multivariate skew- t distribution, called mixtures of skew- t factor analysis (MSTFA) model. This model can be viewed as a powerful tool for model-based clustering of high-dimensional data where observations in each cluster exhibit non-normal features such as heavy-tailed noises and extreme skewness. Missing values may be frequently present due to the incomplete collection of data. A computationally feasible EM-type algorithm is developed to carry out maximum likelihood estimation and create single imputation of possible missing values under a missing at random mechanism. The numbers of factors and mixture components are determined via penalized likelihood criteria. The utility of our proposed methodology is illustrated through analysing both simulated and real datasets. Numerical results are shown to perform favourably compared to existing approaches.

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial distribution, and some discrete distributions arising from the multivariate discrete normal distribution.

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.

The multivariate log-normal distribution is used by many authors and statistical uncertainty propagation programs for inherently positive quantities. Sometimes it is claimed that the log-normal distribution results from the maximum entropy principle, if only means, covariances and inherent positiveness of quantities are known or assumed to be known. In this article we show that this is not true. Assuming a constant prior distribution, the maximum entropy distribution is in fact a truncated multivariate normal distribution – whenever it exists. However, its practical application to multidimensional cases is hindered by lack of a method to compute its location and scale parameters from means and covariances. Therefore, regardless of its theoretical disadvantage, use of other distributions seems to be a practical necessity.

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.

The problem of testing mutual independence of p random vectors in a general setting where the dimensions of the vectors can be different and the distributions can be discrete, continuous or both is of great importance. We propose such a test which utilizes multivariate characteristic functions and is a generalization of known results. We characterize the limiting distribution of the test statistic under the null hypothesis. The limiting null distribution is approximated and the method is validated. Numerical results based on simulations are investigated and our methodology is implemented in the R package IndependenceTests. Power comparisons are also presented for some partial cases of our general test, where some competitive procedures exist.

In this paper, we first discuss some of the well-known methods available in the literature for the estimation of the parameters of a univariate/multivariate stable distribution. Based on the availa...

The problem of estimation of the parameters of a selected univariate distribution has been studied extensively in the literature, in which a one dimensional random parameter is estimated using a random sample of one dimensional observations. There are situations where the selection process is performed using multiple variables or the selection of the population is done using an auxiliary variable. It is natural that such a random vector has a multivariate distribution with a multidimensional parameter space. In this paper, certain methods are developed for estimation of the multidimensional parameters of the selected multivariate distribution. The risks of the proposed estimators are estimated and certain sufficient conditions for inadmissibility of the estimators are given for two classes of the estimators. The results are applied to a prostate cancer data set to illustrate the applicability of theoretical results.

This article proposes the elliptical multivariate leptokurtic-normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram–Charlier expansion of the multivariate normal (MN) distribution and has a closed form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model-based clustering. An EM algorithm is presented to specifically obtain maximum likelihood estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions. The Canadian Journal of Statistics xx: 1–25; 2016 © 2016 Statistical Society of Canada

In this article, by considering a multivariate normal mean–variance mixture distribution, we derive the exact joint distribution of linear combinations of order statistics and their concomitants. From this general result, we then deduce the exact marginal and conditional distributions of order statistics and their concomitants arising from this distribution. We finally illustrate the usefulness of these results by using a Swiss markets dataset.

We study some of the main probabilistic properties of the so called multivariate skew-normal-Cauchy distribution. Simple expressions to compute the entries of the expected Fisher information matrix of this multivariate distribution are proposed.

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.

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component-wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max-stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.

Skew-normal distribution is an extension of the normal distribution where the symmetry of the normal distribution is distorted with an extra parameter. A multivariate skew-normal distribution has been parametrized differently to stress different aspects and constructions behind the distribution. There are several possible parametrizations available to define the skew-normal distribution. The current most common parametrization is through Ω and α , as an alternative, parametrization through Ω and δ can be used if straightforward relation to marginal distributions is of interest. The main problem with { Ω , δ }-parametrization is that the vector δ cannot be chosen independently of Ω . This motivated us to investigate what are the possibilities of choosing δ under different correlation structures of Ω . We also show how the assumptions on structure of δ and Ω affect the asymmetry parameter α and correlation matrix R of corresponding skew-normal random variable.

Univariate Birnbaum-Saunders distribution has received a considerable attention in recent years. Rieck and Nedelman (Technometrics, vol. 33, 51–60, 1991) introduced a log Birnbaum-Saunders distribution. We introduce a multivariate log Birnbaum-Saunders distribution and discuss its different properties. It is observed that the proposed multivariate model can be obtained from the multivariate Gaussian copula. We have proposed the maximum likelihood estimators of the unknown parameters. Since it is a computationally challenging problem, particularly if the dimension is high, we have considered the approximate maximum likelihood estimators based on the Copula structure using two-step procedure. The asymptotic distributions of both these estimators have been obtained. We compare their performances using Monte Carlo simulations, and it is observed that their performances are very similar in nature. One data set has been analyzed for illustrative purposes.

In the univariate setting, the coefficient of variation is widely used to measure the relative dispersion of a random variable with respect to its mean. Several extensions of the univariate coefficient of variation to the multivariate setting have been introduced in the literature. In this paper, we focus on a distance-based multivariate coefficient of variation. First, some real examples are discussed to motivate the use of the considered multivariate dispersion measure. Then, the asymptotic distribution of several estimators is analyzed under elliptical symmetry and used to construct approximate parametric confidence intervals that are compared with non-parametric intervals in a simulation study. Under normality, the exact distribution of the classical estimator is derived. As this natural estimator is biased, some bias corrections are proposed and compared by means of simulations.

We propose a multivariate functional linear regression (mFLR) approach to analysis and prediction of multivariate functional data in cases in which both the response and predictor variables contain multivariate random functions. The mFLR model, coupled with the multivariate functional principal component analysis approach, takes the advantage of cross-correlation between component functions within the multivariate response and predictor variables, respectively. The estimate of the matrix of bivariate regression functions is consistent in the sense of the multi-dimensional Gram-Schmidt norm and is asymptotically normally distributed. The prediction intervals of the multivariate random trajectories are available for predictive inference. We show the finite sample performance of mFLR by a simulation study and illustrate the method through predicting multivariate traffic flow trajectories for up-to-date and partially observed traffic streams.

In this paper, we obtain some conditions to compare multivariate mixture models with respect to some well-known multivariate stochastic orders. We also utilize the established results in reliability theory to compare the vectors of residual life-lengths of live components of ( n − k + 1 ) -out-of- n systems in both one sample and two samples situations.

A unified framework to analyse multivariate kernel estimators of distribution and survival functions is introduced, before turning our attention to receiver operating characteristic (ROC) curves. These are well-established visual analytic tools for univariate data samples, though their generalisation to multivariate data has been limited. Since non-parametric multivariate kernel smoothing methods possess excellent visualisation properties, they serve as a solid basis for their estimation. With optimal data-based bandwidth matrix selectors, we demonstrate that they possess suitable properties for exploratory data analysis of simulated and experimental data.

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.

For (Xt) a two-sided α-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form Xt=(Xt−m,…,Xt,Xt+1,…,Xt+h), m≥0, h≥1, are multivariate alpha-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders -sets {s∈Rm+h+1:∥s∥=1} for ∥⋅∥ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors Xt when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and (Xt) will have to be "anticipative enough" for Xt to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. The approach extends to processes resulting from the linear combination of stable moving averages and applied to several examples.

For (Xt) a two-sided α-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form Xt=(Xt−m,…,Xt,Xt+1,…,Xt+h), m≥0, h≥1, are multivariate alpha-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders -sets {s∈Rm+h+1:∥s∥=1} for ∥⋅∥ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors Xt when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and (Xt) will have to be "anticipative enough" for Xt to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. The approach extends to processes resulting from the linear combination of stable moving averages and applied to several examples.

Resumen El objetivo de esta investigación es proponer un modelo de volatilidad multivariable, el cual combina la propiedad de la distribución α-estable para ajustar colas pesadas con el modelo GARCH para capturar clúster de volatilidad. El supuesto inicial es que los rendimientos siguen una distribución sub-Gaussiana, la cual es un caso particular de las distribuciones estables multivariadas. El modelo GARCH propuesto se aplica en la estimación del VaR a un portafolio compuesto por cinco activos que cotizan en la Bolsa Mexicana de Valores (BMV). En particular, se compara el desempeño del modelo propuesto con la estimación del VaR obtenida bajo la hipótesis multivariada Gaussiana, t-Student y Cauchy durante el período de la crisis financiera de 2008. Abstract The purpose of this investigation is to propose a multivariate volatility model that takes into consideration time varying volatility and the property of the α-stable sub-Gaussian distribution to model heavy tails. The principal assumption is that returns follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. The proposed GARCH model is applied to a Value at Risk (VAR) estimation of a portfolio composed by 5 companies listed in the Mexican Stock Exchange Index (IPC) and compared with the one obtained using the normal multivariate distribution, t-Student and Cauchy. In particular, we examine performances during the financial crisis of 2008.

The multivariate normal distribution was briefly introduced in Chapter 2. In this chapter we consider its properties in some detail since the estimation of its parameters is the source of many standard multivariate statistical methods. In addition, we shall meet a number of derived distributions of fundamental importance.

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.