Power comparisons of two-sided tests of equality of two covariance matrices based on six criteria (original) (raw)

1979, Annals of the Institute of Statistical Mathematics

Power studies of tests of equality of covariance matrices of two p-variate normal populations ~v~=Z2 against two-sided alternatives have been made based on the following six criteria: 1) Roy's largest root, 2) Hotelling's trace, 3) Pillai's trace, 4) Wilks' criterion, 5) Roy's largestsmallest roots and 6) modified likelihood ratio. A general theorem has been proved establishing the local unbiasedness conditions connecting the two critical values for tests 1) to 5). Extensive unbiased power tabulations have been made for 29=2, for various values of n,, n2, 21 and 22 where n~ is the df of the SP matrix from the ith sample and 2~ is the ith latent root of I~/:; ~ (i=1, 2). Further, comparisons of powers of tests 1) to 5) have been made with those of the modified likelihood ratio after obtaining the exact distribution of the latter for n2=2nl and p=2. Equal tail areas approach has also been used further to compute powers of tests 1) to 4) for p = 2 for studying the bias. Again, a separate study has been made to compare the powers of the largest-smallest roots test with its three biased approximate approaches as well as the largest root. Since the largest root test was observed to have some advantage over the others, critical values were also obtained for this test in the unbiased as well as equal tail areas case for p-3. 1. Introduction Let X1 (pxnl) and X2 (p• p<__n~, i = 1 , 2, be independent matrix variates, columns of X1 being independently distributed as N(O, I~) and AMS 1970 subject classifications: 62H10, 62H15.