Non-finite Fisher information and homogeneity: an EM approach

Even simple examples of finite mixture models can fail to fulfil the regularity conditions that are routinely assumed in standard parametric inference problems. Many methods have been investigated for testing for homogeneity in finite mixture models, for example, but all rely on regularity conditions including the finiteness of the Fisher information and the space of the mixing parameter being a compact subset of some Euclidean space. Very simple examples where such assumptions fail include mixtures of two geometric distributions and two exponential distributions, and, more generally, mixture models in scale distribution families. To overcome these difficulties, we propose and study an <scp>em</scp>-test statistic, which has a simple limiting distribution for examples in this paper. Simulations show that the <scp>em</scp>-test has accurate Type I errors and is more efficient than existing methods when they are applicable. A real example is included.