Revisiting the Hodges-Lehmann estimator in a location mixture model: Is asymptotic normality good enough?

Abstract

Location mixture models, resulting in shifting a common distribution with some probability, have been widely used to account for existence of clusters in the data. Assuming only symmetry of this common distribution allows for great flexibility, especially when the traditional normality assumption is violated. This semi-parametric model has been studied in several papers, where the mixture parameters are first estimated before constructing an estimator for the non-parametric component. The plug-in method suggested by Hunter et al. (2007) has the merit to be easily implementable and fast to compute. However, no result is available on the limit distribution of the obtained estimator, hindering for instance construction of asymptotic confidence intervals. In this paper, we give sufficient conditions on the symmetric distribution for asymptotic normality to hold. In case the symmetric distribution admits a log-concave density, our assumptions are automatically satisfied. The obtained result has to be used with caution in case the mixture location are too close or the mixing probability is close to $0$ or $1$. Three examples are considered where we show that the estimator is not to be advocated when the mixture components are not well separated.