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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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