In the statistical and actuarial literature, -quantiles, , represent an important class of risk measures defined through an asymmetric -power loss function that generalize the classical (-)quantiles. By exploiting inter-order relations between partial moments, we show that for a Student's distribution with degrees of freedom the -quantile and the -quantile always coincide for any . For instance, for a Student's distribution with 4 degrees of freedom, the -quantile and -quantile are equal and the same holds for the -quantile and -quantile; for this distribution, closed form expressions for the -quantile, are provided. Explicit formulas for the central moments are also established. The usefulness of exact formulas is illustrated on real-world financial data.

中文翻译：

---

Student t 分布的 L 分位数等价之间的阶间关系

在统计和精算文献中，-分位数 代表一类重要的风险度量，通过推广经典 (-) 分位数的不对称 - 幂损失函数定义。通过利用部分矩之间的阶间关系，我们表明对于具有自由度的学生分布，对于任何 ， - 分位数和 - 分位数总是一致的。例如，对于具有 4 个自由度的学生分布，-分位数和 -分位数相等，并且同样适用于 -分位数和 -分位数；对于此分布，提供了分位数的封闭形式表达式。还建立了中心矩的显式公式。真实世界的金融数据说明了精确公式的有用性。