Variable selection and semiparametric efficient estimation for the heteroscedastic partially linear single-index model

An efficient estimating equations procedure is developed for performing variable selection and defining semiparametric efficient estimates simultaneously for the heteroscedastic partially linear single-index model. The estimating equations are proposed based on the smooth threshold estimating equations by using the efficient score function of partially linear single-index models. And this estimating equations procedure can be used to perform variable selection without solving any convex optimization problems, and automatically eliminate nonsignificant variables by setting their coefficients as zero. The resulting estimators enjoy the oracle property and are semiparametrically efficient. The finite sample properties of the proposed estimators are illustrated by some simulation examples, as well as a real data application.