Penalized minimum-distance estimates in finite mixture models

When finite mixture models are used to fit data, it is sometimes important to estimate the number of mixture components. A nonparametric maximum-likelihood approach may result in too many support points and, in general, does not yield a consistent estimator. A penalized likelihood approach tends to produce a fit with fewer components, but it is not known whether that approach produces a consistent estimate of the number of mixture components. We suggest the use of a penalized minimum-distance method. It is shown that the estimator obtained is consistent for both the mixing distribution and the number of mixture components.