Properties of convergence of a fuzzy set estimator of the density function

A Criterion for the Fuzzy Set Estimation of the Regression Function

“…This absence of functional characteristics complicates the evaluation of the estimator θ n using a sample, as well as the evaluation of the fuzzy set estimator of the regression function if it is defined in terms of θ n . The method of fuzzy set estimation of the regression function introduced by Fajardo et al 3 is based on defining a fuzzy set estimator of the Nadaraya-Watson type, for independent pairs of data, in terms of the fuzzy set estimator of the density function introduced in Fajardo et al 4 . Moreover, the regression function is estimated by means of an average fuzzy set estimator considering pairs of fixed data, which is a particular case if we consider independent pairs of nonfixed data.…”

“…In this paper we estimate the regression function by means of the nonparametric and fuzzy set estimator of the Nadaraya-Watson type, for independent pairs of data, introduced by Fajardo et al 3 , obtaining a significant reduction of the integrated mean square error of the fuzzy set estimator regarding the integrated mean square error of the classic kernel estimators. This reduction is obtained by the conditions imposed on the thinning function, a function that allows to define the estimator proposed by Fajardo et al 4 , which implies that the fuzzy set estimator has better performance than the kernel estimations. The above reduction is not obtained in Fajardo et al 3 .…”