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A bootstrap-based method to achieve optimality in estimating the extreme-value index. (English) Zbl 0972.62014

Let \(\xi\) be a random variable whose distribution function \(F\) is in the domain of attraction of an extreme-value distribution \(G_\gamma\). Suppose that we have i.i.d. observations \(X_1, X_2, \ldots, X_n\), where \(X_i,\;i=1, \ldots, n\), have the distribution function \(F.\) The problem of estimation of the shape parameter \(\gamma\) is considered in this paper. There exist two estimators. Pickands’ estimator: \[ \gamma_{n,\theta}(k)=-(\log\theta)^{-1}\log\{(X_{n,n-[k\theta^2]}-X_{n,n-[k\theta]})(X_{n,n-[k\theta]} - X_{n,n-k})^{-1}\}, \] where \(\theta \in (0, 1)\) and \(X_{n,k}\) are the order statistics of \(X_1, X_2, \ldots, X_n.\) And the moment estimator: \[ \gamma_{n,2} (k)=M_n^{(1)}(k)+1-2^{-1}[(1-(M_n^{(1)}(k))^{2}/M_n^{(2)}(k)]^{-1}, \] where \(M_n^{(j)} (k)=k^{-1}\sum_{i=0}^{k-1} (\log X_{n,n-j} - \log X_{n,n-k})^j.\) These estimators are consistent for \(\gamma\) if \(k = k(n) \to \infty\), \(k(n) =o(n)\), \(n\to \infty.\) If one increases the speed at which \(k(n)\) goes to infinity, the asymptotic variance decreases, but the asymptotic bias increases.
There is an optimal sequence balancing variance and bias components. The authors propose a procedure for obtaining the optimal sequence \(k_0(n)\), where they assume a second order expansion, but do not assume the second order (or first order) characteristic known. This procedure is based on a double bootstrap method. Results are presented for the Pickands estimator and for the moment estimator. Application of the procedure to North Sea wave height data are considered.

MSC:

62F12 Asymptotic properties of parametric estimators
62F40 Bootstrap, jackknife and other resampling methods
62G32 Statistics of extreme values; tail inference
62P12 Applications of statistics to environmental and related topics
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