A Robbins-Monro procedure for estimation in semiparametric regression models

TL;DR

This paper introduces an efficient Robbins-Monro based method for simultaneous parametric shift estimation and nonparametric regression function estimation in semiparametric models, with proven convergence and normality.

Contribution

It presents a novel combined stochastic algorithm for shift and regression function estimation that does not require preliminary regression evaluation.

Findings

Almost sure convergence of estimators

Asymptotic normality of estimates

Effective application on simulated and real data

Abstract

This paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model. We implement a very efficient and easy to handle Robbins-Monro procedure. On the one hand, we propose a stochastic algorithm similar to that of Robbins-Monro in order to estimate the shift parameter. A preliminary evaluation of the regression function is not necessary to estimate the shift parameter. On the other hand, we make use of a recursive Nadaraya-Watson estimator for the estimation of the regression function. This kernel estimator takes into account the previous estimation of the shift parameter. We establish the almost sure convergence for both Robbins-Monro and Nadaraya--Watson estimators. The asymptotic normality of our estimates is also provided. Finally, we illustrate our semiparametric estimation procedure on simulated and real data.