A Robbins-Monro procedure for estimation in semiparametric regression models

Bernard Bercu (1, 2) , Philippe Fraysse (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 ALEA - Advanced Learning Evolutionary Algorithms

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 Robbins-Monro procedure very efficient and easy to handle. 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 for estimating 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 in 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.

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Bernard Bercu, Philippe Fraysse. A Robbins-Monro procedure for estimation in semiparametric regression models. Annals of Statistics, 2012, 40 (2), pp.666-693. ⟨10.1214/12-AOS969⟩. ⟨hal-00551832⟩