THE DISTRIBUTION OF THE LIKELIHOOD RATIO FOR MIXTURES OF DENSITIES FROM THE ONE-PARAMETER EXPONENTIAL FAMILY

We here consider.testing the hypothesis of homogeneity against the alt ernative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log lambda(n), where lambda(n) is the likelihood ratio statistic. The main result, obtained by simula tion, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specifi c (differs if the model is changed from Poisson to normal, say), and i s not at all well approximated by the conventional chi(2)2-distributio n obtained by counting parameters. In Section 3, the binomial with sam ple size parameter 2 is considered. Via a simple geometric characteriz ation the case for which the likelihood ratio is 1 can easily be ident ified and the corresponding probability is found. Closed form expressi ons for the likelihood ratio lambda(n) are possible and the asymptotic distribution of 2 log lambda(n) is shown to be the mixture giving equ al weights to the one point distribution with all its mass equal to ze ro and the chi2-distribution with 1 degree of freedom. A similar resul t is reached in Section 4 for the Poisson with a small parameter value (theta less-than-or-equal-to 0.1), although the geometric characteriz ation 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 lambda(n) are found by simulation for various populations and shown to be nearly independent of the population parameter, and appro ximately equal to the (1 - 2alpha)100 percentiles chi(1)2. In Sections 6 and 7, we close with a study of two continuous densities, the expon ential and the normal with known variance. In these models the asympto tic distribution of 2 log lambda(n) is pivotal. Selected (1 - alpha)10 0 percentiles are presented and shown to differ between the two models .