Minimum Hellinger distance estimation in a two-sample semiparametric model

Abstract

We investigate the estimation problem of parameters in a two-sample semiparametric model. Specifically, let X 1 , … , X n be a sample from a population with distribution function G and density function g . Independent of the X i ’s, let Z 1 , … , Z m be another random sample with distribution function H and density function h ( x ) = exp [ α + r ( x ) β ] g ( x ) , where α and β are unknown parameters of interest and g is an unknown density. This model has wide applications in logistic discriminant analysis, case-control studies, and analysis of receiver operating characteristic curves. Furthermore, it can be considered as a biased sampling model with weight function depending on unknown parameters. In this paper, we construct minimum Hellinger distance estimators of α and β . The proposed estimators are chosen to minimize the Hellinger distance between a semiparametric model and a nonparametric density estimator. Theoretical properties such as the existence, strong consistency and asymptotic normality are investigated. Robustness of proposed estimators is also examined using a Monte Carlo study.