Variance and Distribution of the Graybill-Deal Estimator of the Common Mean of Two Normal Populations
两个正态总体共同均值的 Graybill–Deal 估计量方差与分布
Let $x$ and $y$ be two independent normal variables with mean $\mu$ and variances $\sigma^2_1$ and $\sigma^2_2$ respectively. Also let $S^2_1$ and $S^2_2$ be two independent estimators of $\sigma^2_1$ and $\sigma^2_2$ such that $mS^2_1\sigma^{-2}_1$ and $nS^2_2\sigma^{-2}_2$ are chi-squares with $m$ and $n$ degrees of freedom respectively. The Graybill-Deal estimator of $\mu$ is $\hat{\mu} = (S^{-2}_1x + S^{-2}_2y)/(S^{-2}_1 + S^{-2}_2)$. In this paper an expression for the…