MAXIMUM SMOOTHED LIKELIHOOD DENSITY ESTIMATION IN TWO-SAMPLE MIXTURE DATA WITH LIKELIHOOD RATIO ORDERING

YU JIADONG

Publication

MAXIMUM SMOOTHED LIKELIHOOD DENSITY ESTIMATION IN TWO-SAMPLE MIXTURE DATA WITH LIKELIHOOD RATIO ORDERING

YU JIADONG

Abstract

In modern scientific research, data with mixture structure have been frequently identified in various research areas, such as finance and psychology. One important feature is that observations come from several subpopulations with membership unknown; this introduces challenges in subsequent analysis. In this thesis, we consider the density estimation for two-sample mixture data, where we assume that for each observation, the probabilities that it belongs to the subpopulations are known, and the subpopulations satisfy likelihood ratio ordering condition. This thesis contains two main parts. In the first part, with smoothed likelihood principal, we propose a kernel-based nonparametric method and derive a majorization-minimization algorithm to compute the estimates. The bandwidth selection can be adaptively incorporated into the algorithm. We show that starting from any initial value, the proposed algorithm converges; and we establish its asymptotic convergence rate. We conduct vast simulation studies to compare the proposed method with existing methods in the literature. In the second part, we propose two applications of our proposed method: (1) receiver operating characteristic (ROC) curve estimation; (2) the estimation of posterior probabilities in binary response data, and its application to category classification. In both applications, the performances of our methods are promising.