Two component semiparametric density mixture models with a known component

Finite mixture models have been successfully used in many applications, such as classification, clustering, and many others. As opposed to classical parametric mixture models, nonparametric and semiparametric mixture models often provide more flexible approaches to the description of inhomogeneous populations. As an example, in the last decade a particular two-component semiparametric density mixture model with a known component has attracted substantial research interest. Our thesis provides an innovative way of estimation for this model based on minimization of a smoothed objective functional, conceptually similar to the log-likelihood. The minimization is performed with the help of an EM-like algorithm.  We show that the algorithm is convergent and the minimizers of the objective functional, viewed as estimators of the model parameters, are consistent. 

More specifically, in our thesis, a semiparametric mixture of two density functions is considered where one of them is known while the weight and the other function are unknown. For the first part, a new sufficient identifiability condition for this model is derived, and a specific class of distributions describing the unknown component is given for which this condition is mostly satisfied. A novel approach to estimation of this model is derived. That approach is based on an idea of using a smoothed likelihood-like functional as an objective functional in order to avoid ill-posedness of the original problem. Minimization of this functional is performed using an iterative Majorization-Minimization (MM) algorithm that estimates all of the unknown parts of the model. The algorithm possesses a descent property with respect to the objective functional. Moreover, we show that the algorithm converges even when the unknown density is not defined on a compact interval. Later, we also study properties of the minimizers of this functional viewed as estimators of the mixture model parameters. Their convergence to the true solution with respect to a bandwidth parameter is justified by reconsidering in the framework of Tikhonov-type functional. They also turn out to be large-sample consistent; this is justified using empirical minimization approach. The third part of the thesis contains a series of simulation studies, comparison with another method and a real data example. All of them show the good performance of the proposed algorithm in recovering unknown components from data.