Estimation of the adjusted standard‐deviatile for extreme risks

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 for independent and identically distributed observations and for β‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.