Bootstrap of the Hill estimator: limit theorems

Published in Annales ISUP, 2010

S. Bouzebda

We develop a bootstrap method for estimating the Pareto index of an extreme value distribution. We begin by considering a sequence \(X_1,\ldots,X_n\) of i.i.d. random variables with distribution function \(F(\cdot)\) satisfying \(\lim_{x\rightarrow \infty}\frac{1-F(xt)}{1-F(x)}=t^{-\frac{1}{c}}~~\mbox{for all}~~ t>0.\)
We establish asymptotic normality properties for the bootstrapped sums of extreme values as well as for the bootstrapped Hill estimator. These results are obtained using weighted approximations of the uniform empirical and quantile processes by suitable Brownian bridges.

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