Estimation of the Adjusted Standard-deviatile for Extreme Risks

TL;DR

This paper introduces the adjusted standard-deviatile, a new risk measure for extreme risks, along with efficient estimation methods and theoretical properties, validated through simulations and real data analysis.

Contribution

It proposes the adjusted standard-deviatile, modifies the Bayes risk for expectiles, and develops asymptotic and estimation techniques for extreme risk assessment.

Findings

Asymptotic expansions derived for the adjusted standard-deviatile.

Two efficient estimators proposed and their asymptotic normality proved.

Simulation and real data show the estimators' good performance.

Abstract

In this paper, we modify the Bayes risk for the expectile, the so-called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard-deviatile. First, we derive the asymptotic expansions of the adjusted standard-deviatile. Next, based on the first-order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard-deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators. Simulations and real data applications are conducted to examine the performance of the proposed estimators.