On the asymptotic behaviour of the recursive Nadaraya–Watson estimator associated with the recursive sliced inverse regression method

Abstract

We investigate the asymptotic behaviour of the recursive Nadaraya–Watson estimator for the estimation of the regression function in a semiparametric regression model. On the one hand, we make use of the recursive version of the sliced inverse regression method for the estimation of the unknown parameter of the model. On the other hand, we implement a recursive Nadaraya–Watson procedure for the estimation of the regression function which takes into account the previous estimation of the parameter of the semiparametric regression model. We establish the almost sure convergence as well as the asymptotic normality for our Nadaraya–Watson estimate. We also illustrate our semiparametric estimation procedure on simulated data.

Keywords:

• semi-parametric regression
• recursive estimation
• Nadaraya–Watson estimator
• sliced inversion regression

2000 Mathematics Subject Classification::

• Primary: 62H12
• Secondary: 62G05
• 60F05
• 62L12

Acknowledgements

The authors thank the Editor, the Associate Editor and the two anonymous referees for their helpful remarks and their valuable comments and suggestions which greatly improved the paper.