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 ξ_{1}, ξ_{2} and unknown covariance matrices Σ_{1}, Σ_{2} (> 0) respectively. We shall show that Pillai’s test, which is locally best invariant, is locally minimax for testing H_{0} : Σ_{1} = Σ_{2} against the alternative H_{1} : tr(Σ_{2}^{−1}Σ_{1} − I) = σ > 0 as σ → 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.