Journal of Statistics: Advances in Theory and ApplicationsAuthor's: Jesús Fajardo, Ricardo Ríos and Luis Rodríguez
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.
