Robustness to normality of a selection rule

Abstract

The robustness to the assumption of normality is considered for a special case of the procedure proposed by Bofinger and Mengersen (1986) for selecting the t best from k populations. These authors found a lower bound on the probability of correct selection, under the assumption of independent normal distributions with common unknown variance and equal numbers of observations from each population. Using Tukeys (1960) Generalised Lambda Distribution, it is shown that for most symmetric distributions the bound, as calculated for the normal distribution, is conservative in the face of nonnormality and in particular for heavier tails. For asymmetric distributions, however, the bound performs badly, although this does not necessarily indicate that the procedure itself is at fault. General methods for assessing the effect of nonnormality for particular datasets are also considered.