Reducing MSE in estimation of heavy tails: a Bayesian approach

TL;DR

This paper introduces a Bayesian approach to tail estimation that reduces bias and controls MSE, improving upon classical methods like Hill by using the extended Pareto distribution and asymptotic analysis.

Contribution

It develops a Bayesian estimator for heavy tail parameters based on the extended Pareto distribution, enhancing bias reduction and MSE control over existing estimators.

Findings

Bayesian estimator's MSE is a weighted average of Hill and ML estimators.

The method shows good MSE performance in simulations.

Appropriate priors are derived for tail probability estimation.

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 Caeiro et al. (2005) minimum variance reduced bias estimators of the Pareto tail index were first proposed 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 second-order parameters. Here we revisit this problem from a Bayesian point of view starting from the extended Pareto distribution (EPD) approximation to excesses over a high threshold, as developed in Beirlant et al. (2009) using maximum likelihood (ML) estimation. Using asymptotic considerations, we derive an appropriate choice of priors leading to a Bayes estimator for which the MSE curve is a weighted average of the Hill and EPD-ML MSE curves for a large range of thresholds, under the same conditions as in Beirlant et al.(2009). A similar result is obtained for tail probability estimation. Simulations show surprisingly good MSE performance with respect to the existing estimators.