Mixture of Regression Models With Varying Mixing Proportions: A Semiparametric Approach

In this article, we study a class of semiparametric mixtures of regression models, in which the regression functions are linear functions of the predictors, but the mixing proportions are smoothing functions of a covariate. We propose a one-step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions. We derive the asymptotic bias and variance of the one-step estimate, and further establish its asymptotic normality. A modified expectation-maximization-type (EM-type) estimation procedure is investigated. We show that the modified EM algorithms preserve the asymptotic ascent property. Numerical simulations are conducted to examine the finite sample performance of the estimation procedures. The proposed methodology is further illustrated via an analysis of a real dataset.