testing group symmetry of a multivariate distribution测试组多元分布的对称性.pdf

Symmetry 2009, 1, 180-200; doi:10.3390/sym1020180 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Testing Group Symmetry of a Multivariate Distribution Lyudmila Sakhanenko Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824-1027, USA; E-mail: luda@, Tel.: +1 517 432-9795, Fax: +1 517 432-1405. Received: 15 September 2009 / Accepted: 24 November 2009 / Published: 26 November 2009 Abstract: We propose and study a general class of tests for group symmetry of a multi- variate distribution, which encompasses different types of symmetry, such as ellipsoidal and permutation symmetries among others. Our approach is based on supremum norms of spe- cial empirical processes combined with bootstrap. We show that these tests are consistent against any fixed alternative. This work generalizes the methodology of Koltchinskii and Sakhanenko [7], developed for ellipsoidal symmetry to the case of group symmetry. It also provides a unified approach to testing different types of symmetry of a multivariate distribu- tion. Keywords: symmetric distributions; Donsker class; linear operator Classification: MSC 62G10, 60F05, 60F99 1. Introduction Let be a compact group of linear transformations (operators) from to . A Borel probability measure on is called -symmetric if and only if there exist an affine nonsingular transformation from onto and a -invariant Borel probability measure such that In other words,