Symmetric Multivariate Distributions Research Articles

Censoring outliers in elliptically symmetric multivariate distribution

In this paper, we discuss the problem of censoring outliers in a dataset with a class of complex elliptically symmetric (CES) distribution, which has various engineering applications. To this end, the regularized cost function is formulated by the maximum likelihood (ML) setting. Since the ML estimation of the free outlier subset requires solving a combinatorial problem, an efficient and low complexity algorithm is proposed that guarantees the convergence to a stationary point. Two independent and dependent penalty terms to the selected subset are considered. The conventional generalized inner product (GIP) and the submodular minimization approaches are employed in the derivation of the proposed algorithm in the independent and dependent cases, respectively. The dependent case improves performance by slightly increasing complexity. As a special case for Gaussian assumption and using a non-regularized cost function, the proposed algorithm reduces to the fast minimum covariance determinant (FMCD) algorithm, which has impressive efficiency and received considerable attention in statistics and computer science. Hence, the proposed algorithm extends/generalizes the FMCD to be applicable to the non-Gaussian distributions. The simulation results highlight that the proposed algorithms exhibit satisfactory performance in CES environments.

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.

Hypotheses tests on the skewness parameter in a multivariate generalized hyperbolic distribution

The class of generalized hyperbolic (GH) distributions is generated by a mean-variance mixture of a multivariate Gaussian with a generalized inverse Gaussian (GIG) distribution. This rich family of GH distributions includes some well-known heavy-tailed and symmetric multivariate distributions, including the Normal Inverse Gaussian and some members of the family of scale-mixture of skew-normal distributions. The class of GH distributions has received considerable attention in finance and signal processing applications. In this paper, we propose the likelihood ratio (LR) test to test hypotheses about the skewness parameter of a GH distribution. Due to the complexity of the likelihood function, the EM algorithm is used to find the maximum likelihood estimates both in the complete model and the reduced model. For comparative purposes and due to its simplicity, we also consider the Gradient (G) test. A simulation study shows that the LR and G tests are usually able to achieve the desired significance levels and the testing power increases as the asymmetry increases. The methodology developed in the paper is applied to two real datasets.

Asymptotic theory for statistics based on cumulant vectors with applications

AbstractFor any given multivariate distribution, explicit formulae for the asymptotic covariances of cumulant vectors of the third and the fourth order are provided here. General expressions for cumulants of elliptically symmetric multivariate distributions are also provided. Utilizing these formulae one can extend several results currently available in the literature, as well as obtain practically useful expressions in terms of population cumulants, and computational formulae in terms of commutator matrices. Results are provided for both symmetric and asymmetric distributions, when the required moments exist. New measures of skewness and kurtosis based on distinct elements are discussed, and other applications to independent component analysis and testing are considered.

Scatter halfspace depth for [formula omitted]-symmetric distributions

We provide exact expressions for the scatter halfspace depth of a rich class of symmetric multivariate distributions, and discuss their properties. As special cases, we recover some results of Chen et al. (2018) and Paindaveine and Van Bever (2018), using simpler proof techniques.

Evaluation of outlier detection method performance in symmetric multivariate distributions

Determining outliers is more complicated in multivariate data sets than it is in univariate cases. The aim of this study is to evaluate the blocked adaptive computationally efficient outlier nominators (BACON) algorithm, the fast minimum covariance determinant (FAST-MCD) method, and the robust Mahalanobis distance (RM) method in multivariate data sets. For this purpose, outlier detection methods were compared for multivariate normal, Laplace, and Cauchy distributions with different sample sizes and numbers of variables. False-negative and false-positive ratios were used to evaluate the methods’ performance. The results of this work indicate that the performance of these methods varies according to the distribution type.

Heteroscedasticity and/or autocorrelation checks in longitudinal nonlinear models with elliptical and AR(1) errors

The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR(1) errors. The elliptical class includes several symmetric multivariate distributions such as normal, Student-t, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.

On decompositional algorithms for uniform sampling from [formula omitted]-spheres and [formula omitted]-balls

We describe a universal conditional distribution method for uniform sampling from n -spheres and n -balls, based on properties of a family of radially symmetric multivariate distributions. The method provides us with a unifying view on several known algorithms as well as enabling us to construct novel variants. We give a numerical comparison of the known and newly proposed algorithms for dimensions 5, 6 and 7.

A new moment matching algorithm for sampling from partially specified symmetric distributions

A new algorithm is proposed for generating scenarios from a partially specified symmetric multivariate distribution. The algorithm generates samples which match the first two moments exactly, and match the marginal fourth moments approximately, using a semidefinite programming procedure. The performance of the algorithm is illustrated by a numerical example.

A maximum entropy characterization of symmetric Kotz type and Burr multivariate distributions

In this paper a maximum entropy characterization is presented for Kotz type symmetric multivariate distributions as well as for multivariate Burr and Pareto type III distributions. Analytical formulae for the Shannon entropy of these multivariate distributions are also derived.

Self-Consistency and Principal Component Analysis

I examine the self-consistency of a principal component axis; that is, when a distribution is centered about a principal component axis. A principal component axis of a random vector X is self-consistent if each point on the axis corresponds to the mean of X given that X projects orthogonally onto that point. A large class of symmetric multivariate distributions are examined in terms of self-consistency of principal component subspaces. Elliptical distributions are characterized by the preservation of self-consistency of principal component axes after arbitrary linear transformations. A “lack-of-fit” test is proposed that tests for self-consistency of a principal axis. The test is applied to two real datasets.

Self-Consistency and Principal Component Analysis

Abstract I examine the self-consistency of a principal component axis; that is, when a distribution is centered about a principal component axis. A principal component axis of a random vector X is self-consistent if each point on the axis corresponds to the mean of X given that X projects orthogonally onto that point. A large class of symmetric multivariate distributions are examined in terms of self-consistency of principal component subspaces. Elliptical distributions are characterized by the preservation of self-consistency of principal component axes after arbitrary linear transformations. A “lack-of-fit” test is proposed that tests for self-consistency of a principal axis. The test is applied to two real datasets.

Self-Consistent Patterns for Symmetric Multivariate Distributions

The set of k points that optimally represent a distribution in terms of mean squared error have been called principal points (Flury 1990). Principal points are a special case of self-consistent points. Any given set of k distinct points in Rp induce a partition of Rp into Voronoi regions or domains of attraction according to minimal distance. A set of k points are called self-consistent for a distribution if each point equals the conditional mean of the distribution over its respective Voronoi region. For symmetric multivariate distributions, sets of self-consistent points typically form symmetric patterns. This paper investigates the optimality of different symmetric patterns of self-consistent points for symmetric multivariate distributions and in particular for the bivariate normal distribution. These results are applied to the problem of estimating principal points.

Local influence in elliptical linear regression models

Influence diagnostic methods are extended in this paper to elliptical linear models. These include several symmetric multivariate distributions such as the normal, Student t-, Cauchy and logistic distributions, among others. For a particular perturbation scheme and for the likelihood displacement the diagnostics agree with those developed for the normal linear regression model by Cook when the coefficients and the scale parameter are treated separately. This result shows the invariance of the diagnostics with respect to the induced model in the elliptical linear family. However, if the coefficients and the scale parameter are treated jointly we have a different diagnostic for each induced model, which makes this approach helpful for selecting the less sensitive model in the elliptical linear family. An example on the salinity of water is given for illustration.

Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions

The k principal points ξ 1, ..., ξ k of a random vector X are the points that approximate the distribution of X by minimizing the expected squared distance of X to the nearest of the ξ j . A given set of k points y 1, ..., y k partition R p into domains of attraction D 1, ..., D k respectively, where D j ,consists of all points x ∈ R p such that ∥ x − y j ∥ < ∥ x − y l ∥, l ≠ j. If E[ X | X ∈ D j ] = y j for each j, then y 1, ..., y k are k self-consistent points of X (∥·∥ is the Euclidian norm). Principal points are a special case of self-consistent points. Principal points and sell-consistent points are cluster means of a distribution and represent a generalization of the population mean from one to several points. Principal points and self-consistent points are studied for a class of strongly symmetric multivariate distributions. A distribution is strongly symmetric if the distribution of the principal components ( Z 1, ..., Z p )′ is invariant up to sign changes, i.e., ( Z 1, ..., Z p )′ has the same distribution as (± Z 1, ..., ± Z p )′. Elliptical distributions belong to the class of strongly symmetric distributions. Several results are given for principal points and self-consistent points of strongly symmetric multivariate distributions. One result relates self-consistent points to principal component subspaces. Another result provides a sufficient condition for any set of self-consistent points lying on a line to be symmetric to the mean of the distribution.

Symmetric Multivariate and Related Distributions.

Part 1 Preliminaries: construction of symmetric multivariate distributions notation of algebraic entities and characteristics of random quantities the d operator groups and invariance dirichlet distribution problems 1. Part 2 Spherically and elliptically symmetric distributions: introduction and definition marginal distributions, moments and density marginal distributions moments density the relationship between (phi) and f conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log-elliptical distribution additive logistic elliptical distributions complex elliptically symmetric distributions. Part 3 Some subclasses of elliptical distributions: multiuniform distribution the characteristic function moments marginal distribution conditional distributions uniform distribution in the unit sphere discussion symmetric Kotz type distributions definition distribution of R(2) moments multivariate normal distributions the c.f. of Kotz type distributions symmetric multivariate Pearson type VII distributions definition marginal densities conditional distributions moments conditional distributions moments some examples extended Tn family relationships between Ln and Tn families of distributions order statistics mixtures of exponential distributions independence, robustness and characterizations problems V. Part 6 Multivariate Liouville distributions: definitions and properties examples marginal distributions conditional distribution characterizations scale-invariant statistics survival functions inequalities and applications.

On Some Shrinkage Estimators of Multivariate Location

For a continuous and diagonally symmetric multivariate distribution, incorporating the idea of preliminary test estimators, a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and $U$-statistics. In an asymptotic setup, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T2 R, based on Tiku's (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T2 R is one the whole considerably more powerful than the prominenet Hotelling T2 statistics. We also develop a robust statistic T2 D for testing that two multivariate distributions (skew or symmetric) are identical; T2 D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σk)f((x-μk)/σk) and the means and variances of these marginal distributions exist.

Sequential Designs for Spherical Weight Functions

Two papers by Box and Draper (1959, 1963) discussed the selection of first and second order rotatable designs when a spherical region of interest was defined and account had to be taken of possible biases due to the fact that the polynomial model under consideration was inadequate by one order. Here we consider a different but related problem. Suppose the intensity of interest in the factor space is represented, not by a spherical region of interest, but by a symmetric multivariate distribution weight function. (Previously it was assumed that total interest lay uniformly over a spherical region and there was no interest outside it.) We wish to specify the runs of a second order design which can be performed sequentially so that both the first order portion and the full second order design provide protection against biases of one order higher. In particular, we shall consider designs which are either the basic central composite type and consist of a “cube” plus “star” plus center points, or designs which contain an extra star. The extra star is required in some cases to allow certain conditions to be met.