Title: On second-order automatic bias reduction for extreme expectile estimation
Authors: Antoine Usseglio-Carleve - ENSAI (France) [presenting]
Stephane Girard - Inria (France)
Gilles Stupfler - ENSAI - CREST (France)
Abstract: Expectiles induce a law-invariant risk measure that has recently gained popularity in actuarial and financial risk management applications. They are determined by tail expectations and induce a coherent and elicitable risk measure, while quantiles are calculated in terms of tail probabilities only and are not coherent, and the quantile-based Expected Shortfall is not elicitable. Expectiles have therefore been suggested as serious candidates for a standard risk measure. Their estimation in the heavy-tailed framework is not without difficulties; currently available estimators of extreme expectiles are typically biased and hence may show poor finite-sample performance even in fairly large samples. We focus here on the construction of bias-reduced extreme expectile estimators for heavy-tailed distributions. The rationale for our construction hinges on a careful investigation of the asymptotic proportionality relationship between extreme expectiles and their quantile counterparts, as well as of the extrapolation formula motivated by the heavy-tailed context. We accurately quantify and estimate the bias incurred by the use of these relationships when constructing extreme expectile estimators. This motivates the introduction of a class of bias-reduced estimators whose asymptotic properties are rigorously shown, and whose finite-sample properties are assessed on a simulation study and several samples of real data.