Journal of Statistics: Advances in Theory and ApplicationsAuthor's: Jesús Fajardo, Ricardo RÃos and Luis RodrÃguez
Pages: [79] - [112]
Received Date: March 12, 2010
Submitted by:
In this paper, we define a nonparametric and fuzzy set estimator of
the Nadaraya-Watson type regression function for independent pair of
data, and we establish the almost sure, in law, and uniform
convergence over compact subset on 
of the proposed estimator, as a natural
extension of the results obtained by [12]. We use Bernstein type
inequality, properties of local asymptotic normality thinned point
processes, and the Collomb decomposition, as well as
Talagrand’s inequalities for empirical processes,
symmetrization techniques, Rademacher averages, and
Vapnick-Chervonenkis dimensions to obtain these results. In
particular, we obtain a limit distribution, whose asymptotic variance
depends only at the point estimation, this does not hold to the kernel
regression estimators. We also compute the convergence rate of the
optimal scaling factor, which coincides with the convergence rate in
classic kernel estimation. Nevertheless, with this estimator, the
optimal rate of convergence calculated by [23] is not obtained for
independent random copies. To obtain it, we will introduce a new
estimator defined through the average fuzzy set estimators of the
density function, which satisfies the convergence properties of the
proposed estimator as well as the desired properties of a good
estimator. Moreover, the thinned point processes allow us to introduce
a thinning function, which can be used to improve the constants that
define the window size of the estimation, extend in a certain sense
the kernel estimation since, if the thinning function is a density
function, the proposed estimator is equivalent to the Nadaraya-Watson
kernel estimator, and select points of the sample with different
probabilities. In contrast to the kernel estimator, which assigns
equal weight to all points of the sample, we now select points from
the sample with different probabilities.
nonparametric regression estimation, thinned point process, fuzzy set density estimator, Vapnick-Chervonenkis dimensions.
