Testing Multivariate Symmetry

Heathcote, C. R.; Rachev, Svetlozar T.; Cheng, Bei

doi:10.1006/jmva.1995.1046

Cited by 61 publications

(37 citation statements)

References 30 publications

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“…An investigation comparing some tests of bivariate symmetry was done by [19]. Extensions to tests of multivariate symmetry were considered by [20].…”

Section: Tests Of Bivariate Symmetrymentioning

confidence: 99%

On Consistent Nonparametric Statistical Tests of Symmetry Hypotheses

Quessy

2016

Symmetry

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Being able to formally test for symmetry hypotheses is an important topic in many fields, including environmental and physical sciences. In this paper, one concentrates on a large family of nonparametric tests of symmetry based on Cramér-von Mises statistics computed from empirical distribution and characteristic functions. These tests possess the highly desirable property of being universally consistent in the sense that they detect any kind of departure from symmetry as the sample size becomes large. The asymptotic behaviour of these test statistics under symmetry is deduced from the theory of first-order degenerate V-statistics. The issue of computing valid p-values is tackled using the multiplier bootstrap method suitably adapted to V-statistics, yielding elegant, easy-to-compute and quick procedures for testing symmetry. A special focus is put on tests of univariate symmetry, bivariate exchangeability and reflected symmetry; a simulation study indicates the good sampling properties of these tests. Finally, a framework for testing general symmetry hypotheses is introduced.

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“…An investigation comparing some tests of bivariate symmetry was done by [19]. Extensions to tests of multivariate symmetry were considered by [20].…”

Section: Tests Of Bivariate Symmetrymentioning

confidence: 99%

On Consistent Nonparametric Statistical Tests of Symmetry Hypotheses

Quessy

2016

Symmetry

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“…Finally, without elaboration, we mention several sources on testing for central symmetry, angular symmetry, or still other notions of symmetry: [15] using projection pursuit and multivariate location regions, [26] and [29], using projection pursuit and the empirical characteristic function, [22] and [54] using Monte Carlo, and [38] and [47] using graphical methods based on statistical depth functions and multivariate quantile functions.…”

Section: Testing For Symmetrymentioning

confidence: 99%

Multivariate Symmetry and Asymmetry

Serfling

2005

Encyclopedia of Statistical Sciences

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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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“…3 Heathcote, Rachev, and Cheng (1995) introduced two procedures for testing the symmetry of a general multivariate distribution. The first is a multivariate generalization of the test of Csörgö and Heathcote (1987); the second considers specifically multivariate α-stable laws.…”

Section: Tests Based On the Characteristic Symmetry Functionmentioning

confidence: 99%

“…., a symmetry test can be performed by examining departures of E[sin(Y t − µ, τ )] from zero, where the mean vector µ should be estimated as the median or some trimmed average. 4 Thus, the HRC test procedure is based on the process n Heathcote, Rachev, and Cheng [1995] for the choice of the working region). The test of the null hypothesis of symmetry involves the following steps:…”

Section: Tests Based On the Characteristic Symmetry Functionmentioning

confidence: 99%

“…The second symmetry test in Heathcote, Rachev, and Cheng (1995) assumes that the observations are in the domain of attraction of an α-stable law. The procedure is based on the tail estimators of the spectral measure (see also Rachev, Mittnik, and Kim, 1996) and assumes that X j = [x 1j , x 2j ], j = 1, .…”

Section: Tests Based On the Characteristic Symmetry Functionmentioning

confidence: 99%

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