A Note on the Adaptive Estimation of a Multiplicative Separable Regression Function

Christophe Chesneau

doi:10.1155/2014/271303

Abstract

We investigate the estimation of a multiplicative separable regression function from a bidimensional nonparametric regression model with random design. We present a general estimator for this problem and study its mean integrated squared error (MISE) properties. A wavelet version of this estimator is developed. In some situations, we prove that it attains the standard unidimensional rate of convergence under the MISE over Besov balls.

Highlights

We consider the bidimensional nonparametric regression model with random design described as follows
Let (Yi, Ui, Vi)i∈Z be a stochastic process defined on a probability space (Ω, A, P), where
Despite the so-called “curse of dimensionality” coming from the bidimensionality of (1), we prove that our wavelet estimator attains the standard unidimensional rate of convergence under the mean integrated squared error (MISE) over Besov balls

Summary

Motivations

We consider the bidimensional nonparametric regression model with random design described as follows. We adapt our methodology to propose an efficient and adaptive procedure It is based on two wavelet thresholding estimators following the construction studied in Chaubey et al [6]. Despite the so-called “curse of dimensionality” coming from the bidimensionality of (1), we prove that our wavelet estimator attains the standard unidimensional rate of convergence under the MISE over Besov balls (for both the homogeneous and inhomogeneous zones). It completes asymptotic results proved by Linton and Nielsen [1] via nonadaptive kernel methods for the structured nonparametric regression model.

Assumptions and Notations

MISE Result

Adaptive Wavelet Estimation

Proofs