We consider the problem of estimating a mixture proportion using data from two different distributions as well as from a mixture of them. Under the model assumption that the log-likelihood ratio of the two densities is linear in the observations, we develop an empirical likelihood ratio based statistic for constructing confidence intervals for the mixture proportion. Under some regularity conditions, it is shown that this statistic converges to a chi-squared random variable. Simulation results indicate that the performance of this statistic is satisfactory. As a by-product, we give estimators for the two distribution functions. Connections with case-control studies and discrimination analysis are pointed out.
"Empirical likelihood ratio based confidence intervals for mixture proportions." Ann. Statist. 27 (4) 1368 - 1384, August 1999. https://doi.org/10.1214/aos/1017938930