Estimation of the adjusted standard‐deviatile for extreme risks
修改了期望分位数风险度量以更好捕捉极端风险，提出调整标准差分位数，并给出两种高效估计方法，适用于独立同分布和混合时间序列数据。
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 for independent and identically distributed observations and for ‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.