On the Lp-quantiles for the Student t distribution

TL;DR

This paper extends the known coincidence of quantiles and expectiles for Student t distributions to L_p-quantiles with p degrees of freedom, providing new theoretical insights and recursive formulas for moments.

Contribution

It generalizes the equivalence of quantiles and expectiles to L_p-quantiles for Student t distributions with any degrees of freedom, including affine transformations.

Findings

Quantiles and L_p-quantiles coincide for Student t with p degrees of freedom.

Derived recursive equations for truncated moments of Student t distribution.

Extended the result to affine transformations of the distribution.

Abstract

L_p-quantiles represent an important class of generalised quantiles and are defined as the minimisers of an expected asymmetric power function, see Chen (1996). For p=1 and p=2 they correspond respectively to the quantiles and the expectiles. In his paper Koenker (1993) showed that the tau quantile and the tau expectile coincide for every tau in (0,1) for a class of rescaled Student t distributions with two degrees of freedom. Here, we extend this result proving that for the Student t distribution with p degrees of freedom, the tau quantile and the tau L_p-quantile coincide for every tau in (0,1) and the same holds for any affine transformation. Furthermore, we investigate the properties of L_p-quantiles and provide recursive equations for the truncated moments of the Student t distribution.