Smoothing the Hill estimator. (English) Zbl 0873.60021

Summary: For sequences of i.i.d. random variables whose common tail \(1-F\) is regularly varying at infinity with an unknown index \(-\alpha<0\), it is well known that the Hill estimator is consistent for \(\alpha^{-1}\) and usually asymptotically normally distributed. However, because the Hill estimator is a function of \(k=k(n)\), the number of upper order statistics used and which is only subject to the conditions \(k\to \infty\), \(k/n\to 0\), its use in practice is problematic since there are few reliable guidelines about how to choose \(k\). The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of \(k\) decreases and the successful use of the estimator is made less dependent on the choice of \(k\). A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.


60F17 Functional limit theorems; invariance principles
60G70 Extreme value theory; extremal stochastic processes
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