$L_p$-quantile has recently been receiving growing attention in risk management since it has desirable properties as a risk measure and is a generalization of two widely applied risk measures, Value-at-Risk and Expectile. The statistical methodology for $L_p$-quantile is not only feasible but also straightforward to implement as it represents a specific form of M-quantile using $p$-power loss function. In this paper, we introduce the concept of Tail Risk Equivalent Level Transition (TRELT) to capture changes in tail risk when we make a risk transition between two $L_p$-quantiles. TRELT is motivated by PELVE in Li and Wang (2023) but for tail risk. As it remains unknown in theory how this transition works, we investigate the existence, uniqueness, and asymptotic properties of TRELT (as well as dual TRELT) for $L_p$-quantiles. In addition, we study the inference methods for TRELT and extreme $L_p$-quantiles by using this risk transition, which turns out to be a novel extrapolation method in extreme value theory. The asymptotic properties of the proposed estimators are established, and both simulation studies and real data analysis are conducted to demonstrate their empirical performance.

翻译：$L_p$分位数作为风险度量具有理想性质，且是两种广泛应用的风险度量——风险价值（Value-at-Risk）和期望损失（Expectile）的推广，近年来在风险管理领域受到越来越多的关注。$L_p$分位数的统计方法不仅可行，而且易于实现，因为它代表了使用$p$次幂损失函数的M-分位数的一种特定形式。本文引入尾部风险等价水平转换（TRELT）的概念，以捕捉在两个$L_p$分位数之间进行风险转换时尾部风险的变化。TRELT的提出受到Li和Wang（2023）中PELVE的启发，但针对尾部风险。由于理论上尚不清楚这种转换如何运作，我们研究了$L_p$分位数的TRELT（以及对偶TRELT）的存在性、唯一性和渐近性质。此外，我们利用这种风险转换研究了TRELT和极端$L_p$分位数的推断方法，这被证明是极值理论中一种新颖的外推方法。我们建立了所提出估计量的渐近性质，并通过模拟研究和实际数据分析验证了其经验性能。