Some necessary uniform tests for spherical symmetry

While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly to their limiting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional integrals that are not easily evaluated in numerical computation. In this paper, we develop some necessary tests for spherical symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and have simple limiting distributions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and power.

[1]  S. Kotz,et al.  The Meta-elliptical Distributions with Given Marginals , 2002 .

[2]  Mark F. J. Steel,et al.  Robust Bayesian inference in lq-spherical models , 1993 .

[3]  A. Zellner.  Bayesian and Non-Bayesian Analysis of the Regression Model with Multivariate Student- t Error Terms , 1976 .

[4]  G. S. Watson,et al.  Goodness-of-fit tests on a circle. II , 1961 .

[5]  Arjun K. Gupta,et al.  Lp-norm spherical distribution , 1997 .

[6]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[7]  K. Fang,et al.  A characterization of multivariate l1-norm symmetric distributions , 1989 .

[8]  Kai-Tai Fang,et al.  Some Applications of Läuter's Technique in Tests for Spherical Symmetry , 2000 .

[9]  Kai-Tai Fang,et al.  Some families of multivariate symmetric distributions related to exponential distribution , 1988 .

[10]  Calyampudi Radhakrishna Rao,et al.  Multivariate analysis : future directions 2 , 1993 .

[11]  Chunsheng Ma,et al.  Multivariate p-norm symmetric distributions , 1995 .

[12]  Yoshihiro Tashiro,et al.  On methods for generating uniform random points on the surface of a sphere , 1977 .

[13]  Discriminant analysis under elliptical populations , 1994 .

[14]  P. Bentler,et al.  A necessary test of goodness of fit for sphericity , 1993 .

[15]  C. Quesenberry,et al.  Power studies of tests for uniformity, II , 1979 .

[16]  M. Stephens.  Use of the Kolmogorov-Smirnov, Cramer-Von Mises and Related Statistics without Extensive Tables , 1970 .

[17]  Fred J. Hickernell,et al.  Testing multivariate uniformity and its applications , 2001, Math. Comput..

[18]  Yasunori Fujikoshi,et al.  An Asymptotic Expansion for the Distribution of Hotelling'sT2-Statistic under Nonnormality , 1997 .

[19]  Runze Li,et al.  Some Q-Q Probability Plots to Test Spherical and Elliptical Symmetry , 1997 .

[20]  V. Koltchinskii,et al.  Testing for Spherical Symmetry of a Multivariate Distribution , 1998 .

[21]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[22]  O. Marrero.  Robustness of Statistical Tests in the Two‐Sample Location Problem , 1985 .

[23]  T. W. Anderson.  Nonnormal Multivariate Distributions: Inference Based on Elliptically Contoured Distributions , 1992 .

[24]  Arnold Zellner,et al.  Corrigenda: Bayesian and Non-Bayesian Analysis of the Regression Model with Multivariate Student-t Error Terms , 1976 .

[25]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[26]  J. Neyman.  »Smooth test» for goodness of fit , 1937 .

[27]  Takeaki Kariya,et al.  Robust Tests for Spherical Symmetry , 1977 .

[28]  Arnold J. Stromberg,et al.  Number-theoretic Methods in Statistics , 1996 .

[29]  Charles P. Quesenberry,et al.  Power studies of some tests for uniformity , 1977 .

[30]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .