Nonparametric Kernel Estimation Based on Fuzzy Random Variables

Abstract

This paper extends the classical nonparametric curve-fitting methods for fuzzy random variables. The study was conducted in two parts: 1) fuzzy density estimation and 2) nonparametric regression. The classical nonparametric density estimation was first developed based on the kernel method for a given fuzzy random sample at crisp and/or fuzzy point based on crisp or fuzzy bandwidth. The classical bandwidth selection was also extended when the underlying population was normal with known or unknown fuzzy variance. Moreover, the classical nonparametric kernel-based regression model with crisp and/or input or output is extended whenever a given bandwidth is a crisp or fuzzy number. The cross-validation procedure for selecting the optimal value of the (crisp) smoothing parameter is also extended to fit the proposed nonparametric regression model. The large sample properties of the proposed fuzzy estimators were also investigated by some theorems. Several numerical examples including that of real-life data are used to illustrate the proposed methods in curve-fitting estimation with crisp and/or fuzzy information. Moreover, the proposed methods were examined in comparison with some other existing methods, and their effectiveness were clarified via some numerical examples and simulation studies. Both theatrical and numerical results indicated that the nonparametric curve-fitting methods significantly reduced the sum of square errors as well as the spreads of the fuzzy curve-fitting estimation.