  The testing problem for the order of finite mixture models has a long history
and remains an active research topic. Since Ghosh and Sen (1985) revealed the
hard-to-manage asymptotic properties of the likelihood ratio test, there has
been marked progress. 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 paper, 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.
