Tests for Equality of Parameter Matrices in Two Multivariate Linear Models

✍ Scribed by Daan G. Nel

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 240 KB
• Volume: 61
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1997.1661

✦ Synopsis

An approximate degrees of freedom test is suggested for hypotheses of the kind H 0 : C$8 1 M=C$8 2 M in two independent multivariate linear models: Y i =X i 8 i += i , i=1, 2, under the assumption of error matrix variate normality and heteroscedasticity. It is shown for specific vector choices of the matrices C and M that the test reduces to approximate degrees of freedom solutions obtained by Nel (1989), Nel and van der Merwe (1986) and Welch (1947) for simpler models.

1997 Academic Press

1. Introduction

Consider two independent multivariate linear models Y i =X i 8 i += i , i=1, 2, where Y i : N i _p; X i : N i _k are of full rank k, and where it is assumed that the parameter matrices 8 i : k_p are comparable in the sense that they are measuring the same attributes in both models. Assume that = i : N i _p are independently normally distributed N Ni p (0; 7 i I Ni ), where denotes the Kronecker or direct product of matrices (Henderson, Pukelsheim, and Searle, 1983). Under the least squares conditions the maximum likelihood estimators of 8 i are:

The unbiased estimators of 7 i are

In this paper we investigate union intersection methods to test hypotheses of the kind H 0 : C$8 1 M=C$8 2 M, where C$: g_k of rank g and M : p_v of rank v are known matrices.

In Section 2 a method to test such hypotheses is presented. In Section 3, simpler hypotheses of this kind are considered, where the matrices C and article no. MV971661 29 0047-259XÂ97 25.00