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Using shrinkage estimators to reduce bias and MSE in estimation of heavy tails. (English) Zbl 1428.62195

Summary: Bias reduction in tail estimation has received considerable interest in extreme value analysis. Estimation methods that minimize the bias while keeping the mean squared error (MSE) under control, are especially useful when applying classical methods such as the B. M. Hill estimator [Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)]. In the case of heavy tailed distributions, F. Caeiro et al. [REVSTAT 3, No. 2, 113–136 (2005; Zbl 1108.62049)] proposed minimum variance reduced bias estimators of the extreme value index, where the bias is reduced without increasing the variance with respect to the Hill estimator. This method is based on adequate external estimation of a pair of parameters of second order slow variation under a third order condition. Here we revisit this problem exploiting the mathematical fact that the bias tends to 0 with increasing threshold. This leads to shrinkage estimation for the extreme value index, which allows for a penalized likelihood and a Bayesian implementation. This new approach is applied starting from the approximation to excesses over a high threshold using the extended Pareto distribution, as developed in [the first author et al., J. Stat. Plann. Inference 139, No. 8, 2800–2815 (2009; Zbl 1162.62044)]. We present asymptotic results for the resulting shrinkage penalized likelihood estimator of the extreme value index. Finite sample simulation results are proposed both for the penalized likelihood and Bayesian implementation. We then compare with the minimum variance reduced bias estimators.

MSC:

62G32 Statistics of extreme values; tail inference
62F10 Point estimation
62F15 Bayesian inference
62J07 Ridge regression; shrinkage estimators (Lasso)
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