Nonparametric D-test for homogeneity with flexible null and alternative hypotheses

Abstract

When testing for homogeneity in a finite normal mixture, one may compare the fit of a null model to that of an alternative model. While there are many possibilities for null and alternative models, perhaps the most common is assuming one and two mixture components as estimated using (penalized) maximum likelihood. In this paper we study a homogeneity test based on a nonparametric density fit; this immediately supplies the alternative model, and after estimating the mode and curvature at the mode, can generate the null model for comparison. Our procedure, called the nonparametric D-test (NpD test), is intended to allow for uncertainty about the number of mixture components and even the possibility of model misspecification. This test relies on the principle of using $$L^2$$ distance to compare model fits and can be modified through the inclusion of a weight function; generation of the null model from the nonparametric density fit corresponds to an “empirical null hypothesis”. The critical values for the NpD test can be estimated by bootstrapping and regression. Under some assumptions, we establish that the NpD test is consistent. We examine the performance of the NpD test in simulation studies, and an application to data on continuous renal replacement therapy is presented. Suggestions are also provided for adapting the NpD test to a non-normal null model.