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ON AN OPTIMUM TEST OF THE EQUALITY

OF TWO COVARIANCE MATRICES

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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 *xi*_{1}, *xi*_{2} and unknown
covariance
matrices *Sigma*_{1}, *Sigma*_{2} (> 0) respectively. We shall show that Pillai's
test, which is locally best invariant, is locally minimax for testing
*H*_{0} : *Sigma*_{1} = *Sigma*_{2} against the alternative
*H*_{1} : tr(*Sigma*_{2}^{-1} *Sigma*_{1} - 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.

**Source**
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