A note on profile likelihood for exponential tilt mixture models

Suppose that independent observations are drawn from multiple distributions, each of which is a mixture of two component distributions such that their log density ratio satisfies a linear model with a slope parameter and an intercept parameter. Inference for such models has been studied using empirical likelihood, and mixed results have been obtained. The profile empirical likelihood of the slope and intercept has an irregularity at the null hypothesis so that the two component distributions are equal. We derive a profile empirical likelihood and maximum likelihood estimator of the slope alone, and obtain the usual asymptotic properties for the estimator and the likelihood ratio statistic regardless of the null. Furthermore, we show the maximum likelihood estimator of the slope and intercept jointly is consistent and asymptotically normal regardless of the null. At the null, the joint maximum likelihood estimator falls along a straight line through the origin with perfect correlation asymptotically to the first order.