Polar angle tangent vectors follow Cauchy distributions under spherical symmetry

JOURNAL OF MULTIVARIATE ANALYSIS

卷:128

Article

Cacoullos, T.

关键词: Multivariate Cauchy;    Spherical symmetry;    Component ratios distribution;    Angular distribution;    t-statistics;   

DOI  :  10.1016/j.jmva.2014.03.010

来源: Elsevier

【 摘 要 】

Let X = (X-1, ... , X-n)' follow a spherically or elliptically symmetric distribution centered at zero, and Y-i = Xi+1/X-1, Y = (Y-1, ... , Yn-1)'. It is shown that under spherical symmetry Y has a symmetric Cauchy distribution and under elliptical symmetry a general Cauchy distribution. Geometrically, Y is the tangent (or cotangent) vector of the polar angle theta(1). The simple case of one ratio is treated in Arnold and Brockett (1992), Jones (1999, 2008). Moreover, it is shown that root n - 1 cot theta(1) follows the t(n-1) distribution, so that the normal theory distributions of Student's t and correlation coefficient r hold under spherical symmetry. (C) 2014 Elsevier Inc. All rights reserved.

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