Asymptotic normality of an adaptive kernel density estimator for finite mixture models

Choice of an appropriate kernel density estimator is a difficult one in minimum distance estimation based on density functions. Particularly, for mixture models, the choice of bandwidth is very crucial because the component densities may have different scale parameters, which in turn necessitate varying amount of smoothing. Adaptive kernel density estimators use different bandwidths for different components, which make them an ideal choice for minimum distance estimation in mixture models. Cutler and Cordero-Braña (1996. Minimum Hellinger distance estimates for parametric models. J. Amer. Stat. Assoc. 91, 1716-1721) introduced such an adaptive kernel density estimator in their work on minimum Hellinger distance estimation of mixture parameters. In this paper, we study a general version of their adaptive kernel density estimator and establish the asymptotic normality of the proposed estimator. We also illustrate the performance of our estimator via a small simulation study.