Approximation to the expectation of a function of order statistics and its applications

Let {U_n,n1} be a sequence of i.i.d. random variables uniformly distributed on the interval (0,1). For eachn1, denote the order statistics ofU₁,,U_n byU_n,1U_n,n. Under very general conditions on the ranksk_n,1,,k_n,m, we give an approximation to for anym-dimensional bounded and Borel measurable functiong. The approximation provides a unified approach to asymptotic distributions of a wide variety of order statistics, including the trimmed sums,U-statistics based on trimmed samples and the estimators for the index of an extreme value distribution. As the first applications, we discuss the central limit theorems for the so-called generalized Pickands' estimator and generalized moment estimator.This project is supported by the National Natural Science Foundation of China and Doctoral Program Foundation of Higher Education.