**Sankhya:
The Indian Journal of Statistics**

2007, Volume 69, Pt. 4, 700--716

**More Powerful Tests for Homogeneity of Multivariate
Normal Mean Vectors under an Order Restriction**

Shoichi Sasabuchi, Kyushu University, Japan

SUMMARY. Consider the problem of testing the homogeneity of several p-variate normal mean vectors under an order restriction. This is a multivariate extension of Bartholomew's ({\it Biometrika}, 1959) problem. When the covariance matrices are known, this problem has been studied to some extent, for example, by Sasabuchi, Inutsuka and Kulatunga ({\it Biometrika}, 1983), Sasabuchi, Miura and Oda ({\it JSCS}, 2003) and some others. We are interested in the case when the covariance matrices are common but unknown. In this case, Sasabuchi, Tanaka and Tsukamoto ({\it Ann. Statist.}, 2003) proposed a test statistic and studied its upper tail probability under the null hypothesis. In the present paper, we provide some tests, which are more powerful than the above test. We derive some theorems about their null distributions and powers.

*AMS (2000) subject classification. *Primary 62F30; secondary 62F03, 62H15.

*Key words and phrases. *Common unknown covariance matrix, isotonic regression, multivariate isotonic regression, multivariate normal distribution,
similar test.