Pairwise Comparisons Using Trimmed Means or M‐Estimators when Working with Dependent Groups

Abstract

Abstract For J dependent groups, let θ j , j = 1, …, J , be some measure of location associated with the j th group. A common goal is computing confidence intervals for the pairwise differences, θ j — θ k , j < k , such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy‐tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey‐McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one‐step M‐estimator is also considered but found to be less satisfactory.