Lan of Thinned Empirical Processes with an Application to Fuzzy Set Density Estimation

We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability of thinning satisfies . It turns out that the central sequence is determined by the limit of the coefficient of variation of the tangent function. The central sequence depends only on the total number of nonthinned observations if and only if this limit is 1 or –1. In this case under suitable regularity conditions, an asymptotically efficient estimator of the underlying parameter can be based on only. An application to density estimation leads to a fuzzy set density estimator, which is efficient in a parametric model. In a nonparametric setup, it can also outperform the usual kernel density estimator, depending on the values of the density and its second derivative.