The paper explores the concept of the \emph{expectile risk measure} within the framework of the Fundamental Risk Quadrangle (FRQ) theory. According to the FRQ theory, a quadrangle comprises four stochastic functions associated with a random variable: ``error'', ``regret'', ``risk'', and ``deviation''. These functions are interconnected through a stochastic function known as the ``statistic''. Expectile is a risk measure that, similar to VaR (quantile) and CVaR (superquantile), can be employed in risk management. While quadrangles based on VaR and CVaR statistics are well-established and widely used, the paper focuses on the recently proposed quadrangles based on expectile. The aim of this paper is to rigorously examine the properties of these Expectile Quadrangles, with particular emphasis on a quadrangle that encompasses expectile as both a statistic and a measure of risk.

翻译：本文在基本风险四边形（FRQ）理论框架下探讨了“期望分位数风险测度”的概念。根据FRQ理论，一个四边形包含与随机变量相关的四种随机函数：“误差”、“遗憾”、“风险”和“偏离”。这些函数通过称为“统计量”的随机函数相互关联。期望分位数是一种风险测度，类似于VaR（分位数）和CVaR（超分位数），可用于风险管理。尽管基于VaR和CVaR统计量的四边形已得到充分建立并广泛应用，但本文聚焦于近期提出的基于期望分位数的四边形。本文旨在严格审视这些期望四边形（Expectile Quadrangles）的性质，特别关注一种将期望分位数同时作为统计量和风险测度的四边形。