Two-Stage Experiments for Estimating a Common Mean

Abstract

Let $\pi_1, \pi_2$ be two normal populations with common mean $\mu$ and variances $\sigma^2_1, \sigma^2_2$, where the parameter values are unknown. Suppose that it is desired to estimate $\mu$, and that the experimental procedure is to take $m$ observations from each population, compute variance estimates, and then take $n - 2m$ observations from that population with the smaller observed variance, where $n$ has been fixed beforehand. Let $R_n(\theta, m) = V_0^{-1} E(\hat\mu - \mu)^2$ be the risk of the estimator $\hat\mu$, where $V_0 = n^{-1} \min (\sigma^2_1, \sigma^2_2)$ and where $\theta = \sigma^2_2/\sigma^2_1$. For a class of "best" estimators, it is shown in this paper that $\sup_\theta R_n(\theta, m) \rightarrow 1$ as $n \rightarrow \infty$ if and only if $m/n \rightarrow 0$ and $m \rightarrow \infty$ as $n \rightarrow \infty$; that $\min_m \sup_\theta R_n(\theta, m) \sim 1 + Cn^{\frac{1}{3}}$ as $n \rightarrow \infty$; and that the minimax sample size is $m \sim Cn^{\frac{2}{3}}$ as $n \rightarrow \infty$.