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.
In this paper, we analyze the diagonally dominant degree for the Perron complement upon several diagonally dominant cases by using the entries and spectral radius of the original matrix. At the same time, we obtain closure properties for the Perron complement of several diagonally dominant matrices. MSC: Primary 15A47; 15A48; secondary 65U05; 65J10
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