Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing [Sigma]1 = [Sigma]2 are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.Nonnull distributions hypergeometric functions of matrix argument zonal polynomial mixtures of beta functions and gamma functions

Distribution of the likelihood ratio criterion for testing [Sigma] = [Sigma]0, [mu] = [mu]0

The exact null distribution of the likelihood ratio criterion for testing H0: [Sigma] = [Sigma]0 and [mu] = [mu]0 against alternatives H1: [Sigma] [not equal to] [Sigma]0 or [mu] [not equal to] [mu]0 in Np([mu], [Sigma]) has been obtained as (a) a chi-square series and (b) a beta series. Percentage points have been tabulated for p = 2(1) 6, [alpha] = .005, .01, .025, .05, .1, and .25 and various values of sample size N.Exact null distribution likelihood ratio criterion chi-square series beta series test specifying mean vector and covariance matrix percentage points

The distribution of the sphericity test criterion

The exact distribution of Mauchly's sphericity test criterion W = S/[trS/p]p, when S is the sum of product matrix from a sample of size N taken from a p-variate normal population, is obtained using contour integration and methods similar to those of Nair and Box. Tables of percentage points for p = 4(1)10, [alpha] = 0.01 and 0.05, and various values of N (including small) are given and comparisons made with approximate percentage points using methods of Box, Mauchly, Tukey and Wilks, and Davis.Exact distribution sphericity test criterion contour integration factorial series percentage points approximations

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.Asymptotic distributions complex distribution likelihood ratio tests covariance structures

Tests of sphericity of normal distributions and the analysis of repeated measures designs

Sphericity criteria, locally best invariant test, likelihood ratio test, repeated measures design, circularity conditions, Box/Geisser-Greenhouse correction factor,