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 $\beta$-mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.

中文翻译：

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极端风险调整后标准差的估计

在本文中，我们修改了预期的贝叶斯风险，即所谓的变异风险度量，以更好地捕获极端风险。修改后的风险度量称为调整后的标准差。首先，我们推导调整后的标准差的渐近展开式。接下来，基于一阶渐近展开，我们提出了两种有效的中间和极端水平调整标准差估计方法。通过使用极值理论的技术，证明了独立同分布观测值的估计量和独立同分布观测值的估计量的渐近正态性。$\beta$-分别混合时间序列。进行模拟和实际数据应用来检查所提出的估计器的性能。