## 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.

### MSC:

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