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

TL;DR

This paper derives inter-order formulas for moments of Student t distributions and explores their implications for $L_p$-quantiles, revealing conditions under which different $L_p$-quantiles coincide.

Contribution

It introduces new inter-order formulas for Student t moments and applies these to analyze properties of $L_p$-quantiles, including their equality at certain confidence levels.

Findings

Derived formulas linking partial moments of different orders.

Established closed-form expressions for complete moments.

Showed $L_{n-j+1}$- and $L_j$-quantiles coincide at any confidence level.

Abstract

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 $(0, 1)$.