Inter-order relations between moments of a Student \(t\) distribution, with an application to \(L_p\)-quantiles

This paper introduces inter-order formulas for partial and complete moments of a Student \(t\) distribution with \(n\) degrees of freedom. We show how the partial moment of order \(n - j\) about any real value \(m\) can be expressed in terms of the partial moment of order \(j - 1\) for \(j\) in \(\{1,\dots, n \}\). Closed form expressions for the complete moments are also established. We then focus on \(L_p\)-quantiles, which represent a class of generalized quantiles defined through an asymmetric \(p\)-power loss function. Based on the results obtained, we also show that for a Student \(t\) distribution the \(L_{n-j+1}\)-quantile and the \(L_j\)-quantile coincide at any confidence level \(\tau\) in \((0, 1)\).