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

摘要: We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability \({\alpha }_{n}\) thinning satisfies }_n \to _{n \infty } 0,n{\alpha \infty\). It turns out that central sequence is determined by limit coefficient variation tangent function. The depends only total number \({\tau }\left( \right)\) nonthinned observations and this 1 or −1. In case under suitable regularity conditions, an asymptotically efficient estimator underlying parameter can be only. An application to density estimation leads a fuzzy set estimator, which in parametric model. nonparametric setup, it also outperform usual kernel depending values its second derivative.