×

zbMATH — the first resource for mathematics

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 \(\alpha\geq 1\).
MSC:
60E15 Inequalities; stochastic orderings
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Bock, M. E., Diaconis, P., Huffer, F. W. and Perlman, M. D. (1987). Inequalities for linear combinations of gamma random variables.Canad. J. Statist.15387-395. · Zbl 0653.60017
[2] Diaconis, P. and Perlman, M. D. (1990). Bounds for tail probabilities of weighted sums of independent gamma random variables. InTopics in Statistical Dependence(Somerset,PA, 1987).Institute of Mathematical Statistics Lecture Notes—Monograph Series16147-166. IMS, Hayward, CA. · Zbl 0769.60018
[3] Karlin, S. (1968).Total Positivity. Stanford Univ. Press, Stanford, CA. · Zbl 0219.47030
[4] Khaledi, B.-E. and Kochar, S. C. (2004). Ordering convolutions of gamma random variables.Sankhyā66466-473. · Zbl 1192.60046
[5] Kochar, S. and Xu, M. (2011). The tail behavior of the convolutions of gamma random variables.J. Statist. Plann. Inference141418-428. · Zbl 1203.62013
[6] Kochar, S. and Xu, M. (2012). Some unified results on comparing linear combinations of independent gamma random variables.Probab. Engrg. Inform. Sci.26393-404. · Zbl 1264.60018
[7] Marshall, A. W., Olkin, I. and Arnold, B. (2009).Inequalities:Theory of Majorization and Its Applications, 2nd ed. Springer, New York.
[8] Rinott, Y., Scarsini, M. and Yu, Y. (2012). A Colonel Blotto gladiator game.Math. Oper. Res.37574-590. · Zbl 1297.91003
[9] Roosta-Khorasani, F. and Székely, G. J. (2015). Schur properties of convolutions of gamma random variables.Metrika78997-1014. · Zbl 1355.60027
[10] Roosta-Khorasani, F., Székely, G. J. and Ascher, U. M. (2015). Assessing stochastic algorithms for large scale nonlinear least squares problems using extremal probabilities of linear combinations of gamma random variables.SIAM/ASA J. Uncertain. Quantif.361-90. · Zbl 1327.65013
[11] Shaked, M. and Shanthikumar, J. G. (2007).Stochastic Orders. Springer, New York. · Zbl 0806.62009
[12] Székely, G. J. and Bakirov, N. K. (2003). Extremal probabilities for Gaussian quadratic forms.Probab. Theory Related Fields126184-202.
[13] Yu, Y. (2009). Stochastic ordering of exponential family distributions and their mixtures.J. Appl. Probab.46244-254. · Zbl 1161.60308
[14] Yu, Y. (2011). Some stochastic inequalities for weighted sums.Bernoulli171044-1053. · Zbl 1225.60035
[15] Zhao, P. and Balakrishnan, N. (2009). Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables.Statist. Probab. Lett.791717-1723. · Zbl 1170.60013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.