Minimum Hellinger distance estimation in a nonparametric mixture model

In this paper, we investigate the estimation problem of the mixture proportion λ in a nonparametric mixture model of the form λ F ( x ) + ( 1 - λ ) G ( x ) using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G as well as from the mixture distribution λ F + ( 1 - λ ) G are available. We construct a minimum Hellinger distance estimator of λ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ . Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.