  In this article, we propose two classes of semiparametric mixture regression
models with single-index for model based clustering. Unlike many
semiparametric/nonparametric mixture regression models that can only be applied
to low dimensional predictors, the new semiparametric models can easily
incorporate high dimensional predictors into the nonparametric components. The
proposed models are very general, and many of the recently proposed
semiparametric/nonparametric mixture regression models are indeed special cases
of the new models. Backfitting estimates and the corresponding modified EM
algorithms are proposed to achieve optimal convergence rates for both
parametric and nonparametric parts. We establish the identifiability results of
the proposed two models and investigate the asymptotic properties of the
proposed estimation procedures. Simulation studies are conducted to demonstrate
the finite sample performance of the proposed models. An application of NBA
data by new models reveals some new findings.
