On the Unique Crossing Conjecture of Diaconis and Perlman on Convolutions of Gamma Random Variables

TL;DR

This paper investigates a conjecture about the crossing behavior of distribution functions of weighted sums of iid gamma variables, proving it for certain shape parameters and disproving it for others.

Contribution

We disprove the conjecture for gamma shape parameter less than 1 and prove it for shape parameters greater than or equal to 1.

Findings

Disproved the conjecture for α<1

Proved the conjecture for α≥1

Clarified conditions for crossing behavior of gamma sums

Abstract

Diaconis and Perlman (1990) conjecture that the distribution functions of two weighted sums of iid gamma random variables cross exactly once if one weight vector majorizes the other. We disprove this conjecture when the shape parameter of the gamma variates is $\alpha <1$ and prove it when $\alpha\geq 1$.