Graybill-Deal Estimator Research Articles

An estimator of the common mean of two normal populations

In this paper we consider the estimation of the common mean of two normal populations when the variances are unknown. If it is known that one specified variance is smaller than the other, then it is possible to modify the Graybill-Deal estimator in order to obtain a more efficient estimator. One such estimator is proposed by Mehta and Gurland (1969). We prove that this estimator is more efficient than the Graybill-Deal estimator under the condition that one variance is known to be less than the other.

Variance and Distribution of the Graybill-Deal Estimator of the Common Mean of Two Normal Populations

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 variance of $\hat{\mu}$ is given. Also bounds for the distribution of $\hat{\mu}$ are studied.