Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series A, Pt. 2, pp. 328--342

COCHRAN THEOREMS FOR MULTIVARIATE COMPONENTS OF VARIANCE MODELS

By

TONGHUI WANG, New mexico State University

RONGLIN FU, Guangzhou Teachers College

and

CHI SONG WONG, University of Windsor

SUMMARY.  Let Mnxp be the set of all n x p matrix over the real field. For the multivariate normal matrix Y in Mnxp with mean $\mu$ and covariance $\sum_{j=1}^{k} V_j \otimes \sum_j$, the necessary and sufficient conditions under which $''{Y\prime W_{i} Y}_{i=1}^L$ (with nonnegative definite Wi) is a family of independent Wishart $W_p(m_{i}, \sum, \lambda_i)$ random matrices $Y\prime W_i Y''$ is obtained. Te Cochran theorem is extended further to include the case where the set of quadratic forms, ${Q_i(Y)}$, is a family of the matrix second degree polynomials $Q_i(Y)=Y\prime W_i Y+B\prime_i Y+Y\prime B_i+D_i$ with $B_i \in M_{n \times p}$ and  $D_i \in M_{p \times p}$. For illustration, our results are applied to MANOVA models.

AMS (1980) subject classification. Primary 62J10, secondary 62H05, 62H10.

Key words and phrases. Cochran's theorem, generalized inverse, idempotent, inner product, Kronecker product, multivariate normal vector, multivariate components of variance, MANOVA, partial MANOVA, Wishart distribution.

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This article in Mathematical Reviews.