Hill统计量的渐近正态性

ASYMPTOTIC NORMALITY OF HILL'S ESTIMATOR

<正> 的正整数。此后,统计量γ_n的渐近性质被广泛地加以研究。Mason证明了γ_n是弱相容的,同时指出:如果k_n=[n~β],0<β<1,那末γ_n也是γ的强相容估计。[3]和[4]讨论了γ_n的渐近正态性;作为正则变化函数性质的一个应用,[5]中对此亦有讨论。本文将在d.f.F连续这一假定下。

If a d.f.F is continuous and varies regularly,then the Hill's estimator for the regularly varying exponent of the d.f.F is connected with the row-sums of an array of conditional iid random variables. Using this fact,we find out the necessary and sufficient conditions for the Hill's estimator to converge weakly to some distribution function.An example is presenied to show that one of the theorems by Csorgo ant Mason may not be true if some other condi-tions are not imposed on it.We also point out that it is possible for the Hill's estimator to yield something like the Esseen-Berry theorem.