Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates.

We study the heteroscedastic partially linear single-index model with an unspecified error variance function, which allows for high dimensional covariates in both the linear and the single-index components of the mean function. We propose a class of consistent estimators of the parameters by using a proper weighting strategy. An interesting finding is that the linearity condition which is widely assumed in the dimension reduction literature is not necessary for methodological or theoretical development: it contributes only to the simplification of non-optimal consistent estimation. We also find that the performance of the usual weighted least square type of estimators deteriorates when the non-parametric component is badly estimated. However, estimators in our family automatically provide protection against such deterioration, in that the consistency can be achieved even if the baseline non-parametric function is completely misspecified. We further show that the most efficient estimator is a member of this family and can be easily obtained by using non-parametric estimation. Properties of the estimators proposed are presented through theoretical illustration and numerical simulations. An example on gender discrimination is used to demonstrate and to compare the practical performance of the estimators.