COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1895 Likelihood Inference in Some Finite Mixture Models Xiaohong Chen, Maria Ponomareva and Elie Tamer May 2013 Parametric mixture models are commonly used in applied work, especially
empirical economics, where these models are often employed to learn for example about the
proportions of various types in a given population. This paper examines the inference
question on the proportions (mixing probability) in a simple mixture model in the presence
of nuisance parameters when sample size is large. It is well known that likelihood
inference in mixture models is complicated due to 1) lack of point identification, and 2)
parameters (for example, mixing probabilities) whose true value may lie on the boundary of
the parameter space. These issues cause the profiled likelihood ratio (PLR) statistic to
admit asymptotic limits that differ discontinuously 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 in a finite mixture model and
provides possible 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. We also examine small sample inferences and projection methods. |
