%0 Journal Article
%T A Note on the Adaptive Estimation of a Multiplicative Separable Regression Function
%A Christophe Chesneau
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/271303
%X 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. 1. Motivations We consider the bidimensional nonparametric regression model with random design described as follows. Let be a stochastic process defined on a probability space , where is a strictly stationary stochastic process, is a strictly stationary stochastic process with support in , and is an unknown bivariate regression function. It is assumed that , exists, are independent, are independent, and, for any , and are independent. In this study, we focus our attention on the case where is a multiplicative separable regression function: there exist two functions and such that We aim to estimate from the random variables: . This problem is plausible in many practical situations as in utility, production, and cost function applications (see, e.g., Linton and Nielsen [1], Yatchew and Bos [2], Pinske [3], Lewbel and Linton [4], and Jacho-Ch¨¢vez [5]). In this note, we provide a theoretical contribution to the subject by introducing a new general estimation method for . A sharp upper bound for its mean integrated squared error (MISE) is proved. Then 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]. It has the features to be adaptive for a wide class of unknown functions and enjoy nice MISE properties. Further details on wavelet estimators can be found in, for example, Antoniadis [7], Vidakovic [8], and H£¿rdle et al. [9]. 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. The paper is organized as follows. Assumptions on (1) and some notations are introduced in Section 2. Section 3 presents our general MISE result. Section 4 is devoted to our wavelet estimator and its performances in terms of rate of convergence under the MISE over Besov
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/271303/