The Block-Matrix Sphericity Test: Exact and Near-Exact Distributions for the Test Statistic

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

In this work near-exact distributions for the likelihood ratio test (l.r.t.) statistic to test the one sample block-matrix sphericity hypothesis are developed under the assumption of multivariate normality. Using a decomposition of the null hypothesis in two null hypotheses, one for testing the independence of the k groups of variables and the other one for testing the equality of the k block diagonal matrices of the covariance matrix, we are able to derive the expressions of the l.r.t. statistic, its h-th null moment, and the characteristic function (c.f.) of its negative logarithm. The decomposition of the null hypothesis induces a factorization on the c.f. of the negative logarithm of the l.r.t. statistic that enables us to obtain near-exact distributions for the l.r.t. statistic. Numerical studies using a measure based on the exact and approximating c.f.'s are developed. This measure is an upper bound on the distance between the exact and approximating distribution functions, and it is used to assess the performance of the near-exact distributions and to compare these with the Box type asymptotic approximation developed by Chao and Gupta (Commun. Stat. Theory Methods 20:1957-1969, 1991).
Original languageUnknown
Title of host publicationRecent Developments in Modeling and Applications in Statistics
EditorsPaulo Eduardo Oliveira, Maria da Graça Temido, Carla Henriques, Maurizio Vichi
Place of PublicationHeidelberg
PublisherSpringer
Pages169-177
ISBN (Print)978-3-642-32418-5 / 978-3-642-32419-2
Publication statusPublished - 1 Jan 2013

Publication series

NameStudies in Theoretical and Applied Statistics
PublisherSpringer

Cite this

Marques, F. J. G. P., & Coelho, C. M. A. (2013). The Block-Matrix Sphericity Test: Exact and Near-Exact Distributions for the Test Statistic. In P. E. Oliveira, M. D. G. Temido, C. Henriques, & M. Vichi (Eds.), Recent Developments in Modeling and Applications in Statistics (pp. 169-177). (Studies in Theoretical and Applied Statistics). Heidelberg: Springer.