Kernel smoothing methods are widely used in many areas of statistics with great success. In particular, minimum distance procedures heavily depend on kernel density estimators. It has been argued that when estimating mixture parameters in finite mixture models, adaptive kernel density estimators would be preferable over non-adaptive kernel density estimators. Cutler and Cordero-Brana (1996) introduced such an adaptive kernel density estimator for the minimum Hellinger distance estimation in finite mixture models. In this paper, we investigate the convergence properties of a practical version of their adaptive estimator under some regularity conditions, and compare them with those of a non-adaptive estimator. Asymptotic distribution of the adaptive estimator is also derived.
Convergence Properties of an Adaptive Kernel Density Estimator for Finite Mixture Models