New modifications of the Bechhofer method (Q5928937)

The method proposed by \textit{R. E. Bechhofer} [Ann. Math. Stat. 25, 16--39 (1954; Zbl 0055.13003)] for the selection of the largest of \(k\) expectations \(\mu_1,\dots,\mu_k\) of \(k\) independent normal random variables \(X_1,\dots,X_k\) is considered. The strongest assertion the method gives is that the expectation \(\mu_I\), corresponding to the largest observation, is the largest among the expectations. This is a somewhat disappointing statement if the difference between the largest and second largest \(X_i\) is large. The Tukey method and the method proposed by \textit{D. G. Edwards} and \textit{J. C. Hsu} [ J. Am. Stat. Assoc. 78, 965--971 (1983; Zbl 0534.62050)] do not have this disadvantage, but have weaker assertions than the Bechhofer method concerning the difference \(\mu_I-\text{Max}_i\mu_i\) for some observations. Two new modifications are proposed here that give stronger assertions than the Tukey and the Edwards and Hsu methods concerning both \(\mu_I-\text{Max}_i\mu_i\) and \(\mu_I--\text{Max}_{i\neq I}\mu_i\).