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On the unique crossing conjecture of Diaconis and Perlman on convolutions of gamma random variables. (English) Zbl 1382.60045
Summary: P. Diaconis and M. D. Perlman [in: Topics in statistical dependence. Proceedings of a symposium on dependence in statistics and probability, held in Somerset, Pennsylvania, USA, August 1–5, 1987. Hayward, CA: Institute of Mathematical Statistics. 147–166 (1990; Zbl 0769.60018)] 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 \(\alpha\geq 1\).
60E15 Inequalities; stochastic orderings
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