Monotonicity properties of the gamma family of distributions

TL;DR

This paper investigates the monotonicity properties of the gamma distribution's tail probabilities with respect to its shape parameter, revealing new insights into their increasing, decreasing, and non-monotonic behaviors.

Contribution

It extends and refines existing results on the monotonicity of gamma distribution tail probabilities across different parameter ranges.

Findings

$\, ext{P}(X_a - a > c)$ increases with $a$ for $c \\ge 0$.

$\, ext{P}(X_a - a > c)$ decreases with $a$ for $c \\le -1/3$.

$\, ext{P}(X_a - a > c)$ is non-monotonic for $c \\in(-1/3,0)$.

Abstract

For real $a>0$, let $X_a$ denote a random variable with the gamma distribution with parameters $a$ and $1$. Then $\mathsf P(X_a-a>c)$ is increasing in $a$ for each real $c\ge0$; non-increasing in $a$ for each real $c\le-1/3$; and non-monotonic in $a$ for each $c\in(-1/3,0)$. This extends and/or refines certain previously established results.