## The Annals of Statistics

- Ann. Statist.
- Volume 31, Number 5 (2003), 1517-1536.

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

Shoichi Sasabushi, Koji Tanaka, and Takeshi Tsukamoto

#### Abstract

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 Bartholomew's [*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.

#### Article information

**Source**

Ann. Statist., Volume 31, Number 5 (2003), 1517-1536.

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1065705117

**Digital Object Identifier**

doi:10.1214/aos/1065705117

**Mathematical Reviews number (MathSciNet)**

MR2012824

**Zentralblatt MATH identifier**

1065.62111

**Subjects**

Primary: 62F30: Inference under constraints

Secondary: 62F03: Hypothesis testing 62H12: Estimation

**Keywords**

Common but unknown covariance matrices multivariate isotonic regression multivariate normal distribution order restriction testing homogeneity of mean vectors upper tail probability

#### Citation

Sasabushi, Shoichi; Tanaka, Koji; Tsukamoto, Takeshi. Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown. Ann. Statist. 31 (2003), no. 5, 1517--1536. doi:10.1214/aos/1065705117. https://projecteuclid.org/euclid.aos/1065705117