Average Derivative Estimation from Biased Data

We investigate the estimation of the density-weighted average derivative from biased data. An estimator integrating a plug-in approach and wavelet projections is constructed. We prove that it attains the parametric rate of convergence under the mean squared error. 1. Introduction The standard density-weighted average derivative estimation problem is the following. We observe i.i.d. bivariate random variables defined on a probability space . Let be the unknown density function of , and let be the unknown regression function given by The density-weighted average derivative is defined by The estimation of is of interest in some econometric problems, especially in the context of estimation of coefficients in index models (see, e.g., Stoker [1, 2], Powell et al. [3], and H？rdle and Stoker [4]). Among the popular approaches, there are the nonparametric techniques based on kernel estimators (see, e.g., H？rdle and Stoker [4], Powell et al. [3], H？rdle et al. [5], and Stoker [6]) or orthogonal series methods introduced in Rao [7]. Recently, Chesneau et al. [8] have developed an estimator based on a new plug-in approach and a wavelet series method. We refer to Antoniadis [9], H？rdle et al. [10], and Vidakovic [11] for further details about wavelets and their applications in nonparametric statistics. In this paper, we extend this estimation problem to the biased data. It is based on the “biased regression model” which is described as follows. We observe i.i.d. bivariate random variables defined on a probability space with the common density function: where is a known positive function, is the density function of an unobserved bivariate random variable , and (which is an unknown real number). This model has potential applications in biology, economics, and many other fields. Important results on methods and applications can be found in, for example, Ahmad [12], Sk？ld [13], Cristóbal and Alcalá [14], Wu [15], Cristóbal and Alcalá [16], Cristóbal et al. [17], Ojeda et al. [18], Cabrera and Van Keilegom [19], and Chaubey et al. [20]. Wavelet methods related to this model can be found in Chesneau and Shirazi [21], Chaubey et al. [22], and Chaubey and Shirazi [23]. Let be the density of ; that is, , , and let be the unknown regression function given by The density-weighted average derivative is defined by We aim to estimate from . To reach this goal, we adapt the methodology of Chesneau et al. [8] to this problem and develop new technical arguments derived to those developed in Chesneau and Shirazi [21] for the estimation of (4). A new wavelet estimator is thus