Sequential Estimation of the Largest Normal Mean when the Variance is Known

Abstract

Given a procedure $\hat{\theta}^\ast$ for estimating the largest mean $\theta^\ast$ of $k$ normal populations with common known variance, it is desired to choose the common sample size $n$ so that the mean squared error (M.S.E.) of $\hat{\theta}^\ast$ does not exceed a given bound, $r$, regardless of the configuration of values of the $k$ means. Let $\Delta_1 \geqq \cdots \geqq \Delta_k = 0$ be the ordered values of $(\theta^\ast - \theta_i)$, where $\theta_1, \cdots, \theta_k$ are the unknown means. The M.S.E. depends on the $\Delta$'s and the conservative approach chooses a sample size $n^\ast$ to hold M.S.E. $\leqq r$ for all $\Delta$'s. Sequential and multisample procedures are considered which use sample information about the $\Delta$'s to reduce sample sizes. Asymptotic properties of the sample size and M.S.E. of the resulting estimates are developed. Improvements over using $n^\ast$ are possible, but with limitations. The sample size behavior of any $\hat{\theta}^\ast$ depends on the limiting variance of the estimator as all of $\Delta_1, \cdots, \Delta_{k-1}$ become infinite.