Asymptotic Distributions of Test Statistics for Matrices Concerning Elliptical Distributions

SYNOPTIC ABSTRACT

This article presents explicitly the results on the asymptotic distributions of the likelihood ratio test statistic −2 log λ (= n ) when the sample is from the nonnormal populations possessing the first four moments similar to those of an elliptically contoured distribution. The statistics are obtained on the various structures of Σ for one or more populations. In all the situations, the asymptotic distributions of n are either noncentral chi–squares or those of a linear function of two noncentral chi–square variates, when the alternatives are close to the null hypotheses. For other alternatives, we get asymptotic normality of (–F₀)/σ₀ where E() = F₀ + o (1) and V() = σ₀²/n + o(n^−l).

Key Words and Phrases:

• sphericity test
• independent of k sets
• complex structure
• intra class-correlation structure
• equality of covariance matrices for k populations