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 *

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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.

FULL PAPER.

This article in Mathematical Reviews.