THE DECOMPOSITION OF THE BEHRENS-FISHER STATISTIC IN q-DIMENSIONAL COMMON PRINCIPAL COMPONENT SUBMODELS

D. G. NEL^{1} AND I. PIENAAR^{2}

1Department of Mathematical Statistics, University of the OFS, P.O. Box 339, Bloemfontein, 9300, South Africa

2Department of Statistics, Vista University, Bloemfontein, 9300, South Africa

(Received December 11, 1995; revised May 27, 1996)

Abstract. Takemura (1985, Multivariate Analysis VI , ed. P. R. Krishnaiah, 583–597, Elsevier, Amsterdam) presented a decomposition of Hotelling’s T^{2}-statistic into analogues of univariate Student-t variates along the principal component axes of the (pooled) sample covariance matrix. In this paper the idea is extended to the heteroscedastic situation where an analogous decomposition of the Behrens-Fisher statistic is considered when the nature of the heteroscedasticity between the two samples can be described by a common principal component (CPC) model, or more particularly a q-dimensional CPC subspace model.

Key words and phrases: Covariance matrices, heteroscedasticity, iterative procedure, model fitting, randomization testing, Takemura decomposition.