Semiparametric Mixture of Regression Models Under Unimodal Error Distribution

A semiparametric model has the advantage of being more robust than a parametric model and more efficient than a nonparametric model. In this paper, a semiparametric regression mixture model, in which the regression coefficients are specified to be parametric and the common sub-distribution is nonparametric, is proposed and studied. For parameter identifiability, the sub-distribution is assumed to be unimodal. The symmetry condition is also considered, and it is shown that it can be used to reduce estimation variability. The semiparametric maximum likelihood method is used to estimate the model parameters. The estimators are shown to be strongly consistent, the convergence rate is derived, and a weak convergence of the estimated density is established with a rate of n1/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{1/3}$$\end{document} and a Chernoff-type weak limit. Simulation studies are conducted to evaluate the performance of the proposed method, and then, the method is applied to analyze a real data.