Asymptotic Expansions of the Distributions of the Likelihood Ratio Criteria for Covariance Matrix
the characteristic function directly. The asymptotic expansion of the non-null distribution of the LR criterion for sphericity (iii), is obtained by using the differential operator due to Welch [15], and also the limiting non-null distribution of the LR test for the equality of k covariance matrices is derived in a similar way. This method has been shown to be useful in other problems in multivariate analysis by Ito [5], Siotani [11], Okamoto [8], and others. All the limiting non-null distributions of these test criteria are shown to be normal distributions, whereas the limiting distributions under hypothesis are x2-distributions as in Box [2] or Anderson ([1], Chapter 10). It may be interesting to note that the limiting non-null distribution of the likelihood ratio criterion for the multivariate linear hypothesis is noncentral X2, the asymptotic expansion of which was obtained by Sugiura and Fujikoshi [13]. 2. Expansion of the distribution of the criterion for z = lo. Let p X 1 vectors