**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 M_{nxp} be the set of all n x p matrix over the real field. For the multivariate normal matrix Y in M_{nxp} 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 W_{i}) 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.