Homogeneity testing under finite location‐scale mixtures

Abstract

The testing problem for the order of finite mixture models has a long history and remains an active research topic. Since Ghosh & Sen (1985) revealed the hard‐to‐manage asymptotic properties of the likelihood ratio test, many successful alternative approaches have been developed. The most successful attempts include the modified likelihood ratio test and the EM‐test, which lead to neat solutions for finite mixtures of univariate normal distributions, finite mixtures of single‐parameter distributions, and several mixture‐like models. The problem remains challenging, and there is still no generic solution for location‐scale mixtures. In this article, we provide an EM‐test solution for homogeneity for finite mixtures of location‐scale family distributions. This EM‐test has nonstandard limiting distributions, but we are able to find the critical values numerically. We use computer experiments to obtain appropriate values for the tuning parameters. A simulation study shows that the fine‐tuned EM‐test has close to nominal type I errors and very good power properties. Two application examples are included to demonstrate the performance of the EM‐test.