The Block-Matrix Sphericity Test: Exact and Near-Exact Distributions for the Test Statistic
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).
KeywordsNull Hypothesis Gamma Distribution Covariance Matrice Asymptotic Approximation Negative Logarithm
This research was partially financially supported by the Portuguese Foundation for Science and Technology, through Centro de Matemática e Aplicações of Universidade Nova de Lisboa (CMA/FCT/UNL), under the project PEst-OE/MAT/UI0297/2011.
- 1.Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003)Google Scholar
- 2.Box, G.E.P.: A general distribution theory for a class of likelihood criteria. Biometrika 36, 317–346 (1949)Google Scholar
- 3.Chao, C.C., Gupta, A.K.: Testing of homogeneity of diagonal blocks with blockwise independence. Commun. Stat. Theor Methods 20, 1957–1969 (1991)Google Scholar
- 4.Cardeño, L., Nagar, D.K.: Testing block sphericity of a covariance matrix. Divulgaciones Matemáticas 9, 25–34 (2001)Google Scholar
- 5.Coelho, C.A.: The generalized integer gamma distribution – a basis for distributions in multivariate statistics. J. Multivar. Anal. 64, 86–102 (1998)Google Scholar
- 6.Coelho, C.A.: The generalized near-integer gamma distribution: a basis for ’near-exact’ approximations to the distributions of statistics which are the product of an odd number of independent Beta random variables. J. Multivar. Anal. 89, 191–218 (2004)Google Scholar
- 7.Coelho, C.A., Marques F.J.: Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions. Comput. Stat., article in press, (2012) doi: 10.1007/s00180-011-0281-1Google Scholar
- 8.Coelho, C.A., Marques, F.J.: The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics. Lin. Algebra Appl. 430, 2592–2606 (2009)Google Scholar
- 9.Marques, F.J., Coelho, C.A.: A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Statistics. Test 20, 180–203 (2011)Google Scholar