Using Shrinkage Estimators to Reduce Bias and MSE in Estimation of Heavy Tails

Abstract

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 Hill (1975) estimator. In the case of heavy tailed distributions, Caeiro et al. (2005) 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 Beirlant et al. (2009). 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.