On Asymptotic Normality of Hill's Estimator for the Exponent of Regular Variation

It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number $k_n$ of extreme order statistics used to construct it tends to infinity appropriately with the sample size $n.$ As our main result, we derive a general condition which can be used to determine the optimal $k_n$ explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition is simplified under appropriate assumptions and then applied to several examples.