THE DISTRIBUTION OF THE LIKELIHOOD RATIO FOR MIXTURES OF DENSITIES

FROM THE ONE-PARAMETER EXPONENTIAL FAMILY

DANKMAR BÖHNING^{1}, EKKEHART DIETZ^{1}, RAINER SCHAUB^{1}, PETER SCHLATTMANN^{1} AND BRUCE G. LINDSAY^{2}

1Department of Epidemiology, Free University Berlin, Augustastr. 37, 12 203 Berlin, Germany 2Department of Statistics, The Pennsylvania State University, 422 Classroom Building, University Park, PA 16802, U.S.A.

(Received November 16, 1992; revised November 29, 1993)

Abstract. We here consider testing the hypothesis of homogeneity 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 χ^{2}-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, the exponential and the normal 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.

Key words and phrases: Aitken acceleration, boundary problem, mixtures, asymptotic distribution of likelihood ratio.