Score estimation of monotone partially linear index model

Abstract

This paper studies semiparametric estimation of a partially linear single index model with a monotone link function. Our estimator is an extension of the score-type estimator developed by Balabdaoui, Groeneboom and Hendrickx (2019) for the monotone single index model, which profiles out the unknown link function by isotonic regression. An attractive feature of the proposed estimator is that it is free from tuning parameters for nonparametric smoothing. We show that our estimator for the finite-dimensional components is √ nconsistent and asymptotically normal. By introducing an additional smoothing to obtain the efficient score, we propose an asymptotically efficient estimator for the finite-dimensional components. Furthermore, we establish the asymptotic validity of a bootstrap inference method based the score-type estimator, which is also free from tuning parameters. A simulation study illustrates usefulness of the proposed method.