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：本文探讨了基本风险象限（FRQ）理论框架下“期望风险度量”的概念。根据FRQ理论，一个象限由与随机变量相关的四个随机函数组成：“误差”、“遗憾”、“风险”和“偏差”。这些函数通过一个称为“统计量”的随机函数相互关联。期望是一种风险度量，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统计量的象限已得到充分确立并被广泛使用，本文重点关注近期提出的基于期望的象限。本文旨在严格考察这些期望象限的性质，特别强调一个将期望同时作为统计量和风险度量的象限。