Smoothing the Conditional Value-at-Risk based Pickands Estimators

TL;DR

This paper introduces a new class of Pickands estimators incorporating CVaR, which are smoothed for robustness, reduce MSE, and demonstrate stable, favorable performance across various distributions.

Contribution

The paper develops a CVaR-based Pickands estimator with smoothing via a beta measure, improving robustness and reducing MSE compared to existing methods.

Findings

Estimators show reduced MSE and high stability in simulations.

The proposed method outperforms traditional estimators across diverse distributions.

Algorithm effectively determines the optimal smoothing parameter.

Abstract

We incorporate the conditional value-at-risk (CVaR) quantity into a generalized class of Pickands estimators. By introducing CVaR, the newly developed estimators not only retain the desirable properties of consistency, location, and scale invariance inherent to Pickands estimators, but also achieve a reduction in mean squared error (MSE). To address the issue of sensitivity to the choice of the number of top order statistics used for the estimation, and ensure robust estimation, which are crucial in practice, we first propose a beta measure, which is a modified beta density function, to smooth the estimator. Then, we develop an algorithm to approximate the asymptotic mean squared error (AMSE) and determine the optimal beta measure that minimizes AMSE. A simulation study involving a wide range of distributions shows that our estimators have good and highly stable finite-sample performance and compare favorably with the other estimators.