Abstract
Let {U n ,n≥1} be a sequence of i.i.d. random variables uniformly distributed on the interval (0,1). For eachn≥1, denote the order statistics ofU 1,…,U n byU n,1≤…≤U n,n . Under very general conditions on the ranksk n,1,…,k n,m , we give an approximation to\(E_g (U_{n,k_{n,1} } , \cdots ,U_{n,k_{n,m} } )\) 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.
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This project is supported by the National Natural Science Foundation of China and Doctoral Program Foundation of Higher Education.
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Cheng, S. Approximation to the expectation of a function of order statistics and its applications. Acta Mathematicae Applicatae Sinica 13, 71–86 (1997). https://doi.org/10.1007/BF02020483
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DOI: https://doi.org/10.1007/BF02020483