The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family

Abstract

We here consider testing the hypothesis ofhomogeneity against the alternative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log λ n , where λ n is the likelihood ratio statistic. The main result, obtained by simulation, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specific (differs if the model is changed from Poisson to normal, say), and is not at all well approximated by the conventional χ (2) 2 -distribution obtained by counting parameters. In Section 3, the binomial with sample size parameter 2 is considered. Via a simple geometric characterization the case for which the likelihood ratio is 1 can easily be identified and the corresponding probability is found. Closed form expressions for the likelihood ratio λ n are possible and the asymptotic distribution of 2 log λ n is shown to be the mixture giving equal weights to the one point distribution with all its mass equal to zero and the χ2-distribution with 1 degree of freedom. A similar result is reached in Section 4 for the Poisson with a small parameter value (θ≤0.1), although the geometric characterization is different. In Section 5 we consider the Poisson case in full generality. There is still a positive asymptotic probability that the likelihood ratio is 1. The upper precentiles of the null distribution of 2 log λ n are found by simulation for various populations and shown to be nearly independent of the population parameter, and approximately equal to the (1–2α)100 percentiles of χ (1) 2 . In Sections 6 and 7, we close with a study of two continuous densities, theexponential and thenormal with known variance. In these models the asymptotic distribution of 2 log λ n is pivotal. Selected (1−α) 100 percentiles are presented and shown to differ between the two models.