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Minimum Hellinger distance estimation in a nonparametric mixture model. (English) Zbl 1156.62024
Summary: We investigate the estimation problem of the mixture proportion $$\lambda$$ in a nonparametric mixture model of the form $$\lambda F(x)+(1-\lambda )G(x)$$ 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 $$\lambda F+(1-\lambda )G$$ are available. We construct a minimum Hellinger distance estimator of $$\lambda$$ 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 $$\lambda$$. 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.

##### MSC:
 62G05 Nonparametric estimation 62F12 Asymptotic properties of parametric estimators 62G07 Density estimation 65C05 Monte Carlo methods 62F10 Point estimation 62F35 Robustness and adaptive procedures (parametric inference)
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