Over the last two decades, semiparametric mixture model receives increasing attention, simply due to the fact that mixture models arise frequently in real life. In this thesis we consider a semiparametric two-component location-shifted mixture model. We propose to use the minimum Hellinger distance estimator (MHDE) to estimate the two location parameters and the mixing proportion. A MHDE is obtained by minimizing the Hellinger distance between an assumed parametric model and a nonparametric estimation of the model. MHDE was proved to have asymptotic effciency and excellent robustness against small deviations from assumed model. To construct the MHDE,we propose to use a bounded linear operator introduced by Bordes et al. (2006) to estimate the unknown nuisance parameter (an unknown function). To facilitate the calculation of the MHDE,we develop an iterative algorithm and propose a novel initial estimation of the parameters of our interest. To assess the performance of the proposed estimations, we carry out simulation studies and
a real data analysis and compare the results with those of the minimum profile Hellinger distance estimator (MPHDE) proposed by Wu et al. (2017) for the same model. The results show that the MHDE is very competitive with the MPHDE in terms of bias and mean squared error, while the MHDE is on average about 2.7 times computationally faster than the MPHDE. The simulation studies also demonstrate that the proposed initial is much more robust than the one
used in Wu et al. (2017).