Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown.

Suppose that an order restriction is imposed among several p-variate normal mean vectors. We are interested in testing the homogeneity of these mean vectors under this restriction. This problem is a multivariate extension of Bartholomewʹs [Biometrika} 46 (1959) 36-48]. When the covariance matrices are known, this problem has been studied by Sasabuchi, Inutsuka and Kulatunga [Hiroshima Math. J. 22 (1992) 551-560], Sasabuchi, Kulatunga and Saito [Amer. J. Math. Management Sci. 18 (1998) 131-158] and some others. In the present paper, we consider the case when the covariance matrices are common but unknown. We propose a test statistic, study its upper tail probability under the null hypothesis and estimate its critical points.