Using Bootstrap Likelihood Ratios in Finite Mixture Models

SUMMARY Statistical inference using the likelihood ratio statistic for the number of components in a mixture model is complicated when the true number of components is less than that of the proposed model since this represents a non-regular problem: the true parameter is on the boundary of the parameter space and in some cases the true parameter is in a nonidentifiable subset of the parameter space. The maximum likelihood estimator is shown to converge to the subset characterized by the same density function, and connection is made to the bootstrap method proposed by Aitkin and co-workers and McLachlan for testing the number of components in a finite mixture and deriving confidence regions in a finite mixture.