Monotone tail and moment ratio properties of Student's family of distributions

TL;DR

This paper investigates the monotonicity properties of tail functions and moment ratios of Student's distributions, establishing key inequalities and their implications for statistical analysis.

Contribution

It proves that the tail ratio G_q(x)/G_p(x) decreases with x for Student's distributions, revealing new monotonicity properties and related moment inequalities.

Findings

G_q(x)/G_p(x) is decreasing in x>0 for 0<p<q≤∞

G_q(x)<G_p(x) for all p<q and x>0

Corollaries on monotonicity of moments and ratios

Abstract

Let G_p denote the tail function of Student's distribution with p degrees of freedom. It is shown that the ratio G_q(x)/G_p(x) is decreasing in x>0 for any p and q such that 0<p<q\le\infty. Therefore, G_q(x)<G_p(x) for all such p and q and all x>0. Corollaries on the monotonicity of (generalized) moments and ratios thereof are also given.