Boundary estimation with the fuzzy set density estimator

In order to extend the properties of the fuzzy set density estimation method and provide new results related to the nonparametric density estimation problems not based on kernels, this paper analyzes the possible boundary effects, if any, of the fuzzy set density estimator and presents a criterion to remove it. Moreover, we propose a boundary fuzzy set estimator which is defined as a particular class of fuzzy set density estimators, where the bias, variance, mean squared error and function that minimizes the mean squared error of the proposed estimator are given. Finally, these theoretical findings are illustrated through some numerical examples, and with two real data examples. Simulations show that the proposed estimator has better performance at points near 0 in a $$b_{_n}$$ spread neighborhood, when it is compared with the particular boundary kernel estimator of a generalized reflection method for the four shapes of densities considered.