On bootstrapping the number of components in finite mixtures of Poisson distributions

Finite mixture models arise in a natural way in that they are modeling unobserved population heterogeneity. It is assumed that the population consists of an unknown number k of subpopulations with parameters λ₁, ..., λ_k receiving weights p₁, ..., p_k. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic (LRS) does not have a (χ²) limit distribution and therefore it is difficult to use the LRS to test for the number of components. These problems are circumvented by using the nonparametric bootstrap such that the mixture algorithm is applied B times to bootstrap samples obtained from the original sample with replacement. The number of components k is obtained as the mode of the bootstrap distribution of k. This approach is presented using the Times newspaper data and investigated in a simulation study for mixtures of Poisson data.