An Asymptotically Optimal Sequential Procedure for the Estimation of the Largest Mean

Interval estimation of the largest mean of k normal populations (k 1) with a common variance σ² is considered. When σ² is known the optimal fixed-width interval is given so that, to have the probability of coverage uniformly lower bounded by γ (preassigned), the sample size needed is minimized. This optimal interval is unsymmetric for $k > 2$. When σ² is unknown a sequential procedure is proposed and its behavior is studied. It is shown that the confidence interval obtained, which is also unsymmetric for $k > 2$, behaves asymptotically as well as the optimal interval. This represents an improvement of the procedure of symmetric intervals considered by the author previously; the improvement is significant, especially when k is large.