Multivariate Symmetry and Asymmetry

Abstract

Abstract. Univariate symmetry has interesting and diverse forms of generalization to the multivariate case. Here several leading concepts of multivariate symmetry -spherical, elliptical, central and angularare examined and various closely related notions discussed. Methods for testing the hypothesis of symmetry, and approaches for measuring the direction and magnitude of skewness, are reviewed.Keywords and Phrases. Multivariate, Symmetry, Skewness, Asymmetry. AMS Subject Classification.Primary: 62H99; Secondary: 62G99 CONCEPTS OF SYMMETRYThe idea of "symmetry" has served, from ancient times, as a conceptual reference point in art and mathematics and in their diverse applications. In aesthetics it is a principle of order, in mathematics an artifact of geometric structure, in philosophy an abstraction of balance and harmony and perfection, in poetry an intuitive essence of nature and divinity. Weyl [52] has created a delightful and wide-ranging treatment of "symmetry", from bilateral symmetry in Greek sculpture to Kant's metaphysical pondering of the problem of left and right to the description of cystalline structure in nature by modern group theory.Here we focus on the notion of symmetry as it relates to multivariate probability distributions in statistical science. Nevertheless even in this specialized context there are many variations on the theme. One can seek to define useful classes of distributions that extend the multivariate normal distribution. Or one can formulate multivariate generalizations of particular univariate distributions such as the exponential. One can define symmetry in terms of structural properties of the distribution function, or of the characteristic function, or of the density function. One may impose invariance of the distribution of a random vector with respect to specified groups of transformations. A useful introduction to these and other approaches is provided by Fang et al. [24]. Other general sources are [43] and [23].A number of widely used examples of multivariate symmetry conveniently may be expressed in terms of invariance of the distribution of a "centered" random vector X − θ in R d under a suitable family of transformations. In increasing order of generality, these are spherical, elliptical, central, and angular symmetry, all of which reduce to the usual notion of symmetry in the univariate case. Below we provide 1 some perspectives on these and closely related notions of multivariate symmetry. Spherical SymmetryA random vector X has a distribution spherically symmetric about θ if rotation of X about θ does not alter the distribution:for all orthogonal d × d matrices A, where " d = " denotes "equal in distribution". In this case X has a characteristic function of the form e it θ h(t t), t ∈ R d , for some scalar function h(·), and a density, if it exists, of the form g((x − θ) (x − θ)), x ∈ R d , for some nonnegative scalar function g (·).Among spherically symmetric distributions are not only multivariate normal distributions with covariance matrices of form σ 2 I d , but...

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