ON AN OPTIMUM TEST OF THE EQUALITY
OF TWO COVARIANCE MATRICES

N. GIRI

Department of Mathematics and Statistics, University of Montreal,
P.O. Box 6128, Station A, Montreal, Quebec, Canada H3C 3J7

(Received September 13, 1990; revised February 21, 1991)

Abstract.    Let X : p × 1, Y : p × 1 be independently and normally distributed p-vectors with unknown means xi1, xi2 and unknown covariance matrices Sigma1, Sigma2 (> 0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H0 : Sigma1 = Sigma2 against the alternative H1 : tr(Sigma2-1 Sigma1 - I)= sigma > 0 as sigma \to 0. However this test is not of type D among G-invariant tests.

Key words and phrases:    Locally best invariant tests, locally minimax tests, type D critical region.

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