A robust alternative to the Lilliefors test of normality

Abstract

The Lilliefors test of normality is a popular and easy-to-explain method for testing whether a sample comes from a normal distribution. Unfortunately, since it relies on the sample mean and sample standard deviation for estimating the parameters of the normal distribution, the Lilliefors test is quite sensitive to the presence of outliers. Contrarily to what could be expected, the substitution of the estimators of location and scale by robust alternatives still does not suffice for obtaining a robust method when either the number of outliers or the sample size is large. In this paper, we propose an actual robust alternative relying on classical and data-driven trimming techniques. The presented test depends on the choice of a subsetting technique, of which we explore three possibilities, and of one parameter, which models the robustness of the test in the presence of outliers. As expected, the choice of parameter is a delicate issue since the gain in robustness comes at the price of a reduced power against some types of alternatives.