Sankhya: The Indian Journal of Statistics

1993, Volume 55, Series A, Pt. 1, 130-149

LINEAR MODELS IN A GENERAL PARAMETRIC FORM

By

CHI SONG WONG, University of Windsor

SUMMARY. The linear model Y $\sim$N$_n (\mu,\sigma^2\Sigma)$ with a restriction $\mu \epsilon \mu_0$ +S is considered, where S = {$Xb:b\epsilon {\mathcal{R}}^p,$R^\prime b= M^\prime u$+$c$for some u in${\mathcal{R}}^l $} is a flat in the n-dimensional euclidean space$\mathcal{R}$and$\mu_0\epsilon {\mathcal{R}}^n$. An explicit formula is obtained for the best unbiased linear estimator$\widehat{K^\prime b}($Y$)$of$ K ^\prime b$, for the F-test of the hypothesis$ K ^\prime b$=$W^\prime v +a $, and for the simultaneous confidence intervals of the$\gamma^\prime b^'s$, where$\gamma^\prime$ranges through the row space of the matrix$K^\prime$. A necessary and sufficient condition is then obtained under which a linear biased estimator of$ K ^\prime b$is no worse than$\widehat{K^\prime b}($Y$)$. Estimators$\widehat{K^\prime b}($Y$)$,$\widehat{\sigma^2}($Y$)$of both$ K ^\prime b$and$\sigma^2$are obtained by using their desired statistical properties rather than using the least square methods or the inverse matrix partition method. None of the known matrices X,$\Sigma\$, R, M, K and W are required to have full column rank and the design matrix X can be one- or multi-way, complete or incomplete, balanced or not balanced, connected or disconnected.

AMS (1985) subject classification. 62J05

Key words and phrases. Biased and unbiased estimators, generalized inverse, linear regression restrictions, mean square error matrix, orgthogonal projection, simultaneous confidence intervals, testing hypotheses.