Kernel excess mass test for multimodality

Abstract

In this paper we propose a new statistical procedure for testing the multimodality of an underlying distribution. Peter Hall developed an innovative idea of calibrating the null distribution for the excess mass test statistic using the empirical distribution function. We find that the qualitative characteristics of a smooth underlying distribution function on the number of modes is barely preserved in the excess mass functional by the non-smooth empirical distribution function. Instead of the empirical distribution function, we propose to use a kernel distribution function estimator. We derive the limiting distribution of the resulting test statistic under strong unimodality, based on which we apply the calibration idea to the proposed test statistic to obtain a cut-off value. Our numerical study suggests that the calibrated kernel excess mass test has greater power than other existing methods. We also illustrate the use of the proposed method in a case study in astronomy which supports an assumption on a physical property of minor planets in the solar system.