Likelihood Inference in Finite Mixture Models with Applications to Experimental Data∗

Parametric mixture models are commonly used in applied work, especially empirical economics, where these models are commonly used to learn for example about the proportions of various types in a given population. It is well known that likelihood inference in these mixture models is complicated due to 1) lack of point identification, and 2) parameters (like some proportions) whose true value lie on the boundary of the parameter space. These issues cause the profiled likelihood ratio statistic to admit asymptotic limits that differ depending on how the true density of the data approaches the regions of singularities where there is lack of point identification. This lack of uniformity in the asymptotic distribution suggests that confidence intervals based on pointwise asymptotic approximations might lead to faulty inferences. This paper examines this problem in details and provides fixes based on the parametric bootstrap. We examine the performance of this parametric bootstrap in Monte Carlo experiments and apply it to data from Beauty Contest experiments.