On Tests for Equicorrelation Coefficient and the Generalized Variance of a Standard Symmetric Multivariate Normal Distribution.

摘要

It is observed that the standard multivariate normal distribution with equicorrelation coefficient, say rho, plays an important role in applied sciences. Tests for rho are derived. The likelihood ratio test is computationally cumbersome, vacuous against one-sided alternatives with positive probability and the exact distribution of the test statistic is nearly intractable. Alternatively, a test based on the best 'natural' unbiased estimator of rho is proposed. It turns out to be locally most powerful and globally unbiased against one-sided alternatives. The exact null and non-null distributions of the test statistic which are of historical interest are derived and the exact percentage points are available. Large sample approximations are also given. With constrained parameter space, a simple test for rho based on the eigen-values of the sample correlation matrix is proposed. The null and non-null asymptotic distributions of the corresponding test statistics are given and the unbiasedness of the test is studies. Finally, we present the likelihood ratio test and a simple test based on the eigen-values of the sample correlation matrix as tests for the generalized variance after establishing that they can be characterized through tests for rho. (Author)