### 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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