On the unique crossing conjecture of Diaconis and Perlman on convolutions of gamma random variables

Abstract

Diaconis and Perlman [In Topics in Statistical Dependence (Somerset, PA, 1987) (1990) 147–166, IMS] conjecture that the distribution functions of two weighted sums of i.i.d. 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 $\alpha\geq1$.