A remark on the comparison of the sum and the maximum of positive random variables

TL;DR

This paper disproves a conjecture about the relationship between the sum and maximum of i.i.d. half-normal variables, extending the approach to generalized gamma distributions.

Contribution

It provides a counterexample to a recent conjecture and introduces a method applicable to generalized gamma distributions.

Findings

Disproved the conjecture for half-normal variables.

Method extends to generalized gamma distributions.

Highlights differences between sum and maximum behaviors.

Abstract

We disprove a conjecture stated in a recent paper by Arnold and Villasenor concerning the sum and the maximum of independent and identically distributed half-normal random variables. Our method is applicable to generalized gamma distributions.