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Stochastic Ordering of Exponential Family Distributions and Their Mixturesxk

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Yaming Yu
Yaming Yu*
Affiliation:
University of California, Irvine
*
∗Postal address: Department of Statistics, University of California, Irvine, Irvine, CA 92697-1250, USA. Email address: [email protected]
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Abstract

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We investigate stochastic comparisons between exponential family distributions and their mixtures with respect to the usual stochastic order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. A general theorem based on the notion of relative log-concavity is shown to unify various specific results for the Poisson, binomial, negative binomial, and gamma distributions in recent literature. By expressing a convolution of gamma distributions with arbitrary scale and shape parameters as a scale mixture of gamma distributions, we obtain comparison theorems concerning such convolutions that generalize some known results. Analogous results on convolutions of negative binomial distributions are also discussed.

Keywords

Binomial mixtureconvolutiongamma mixturehazard rate orderlikelihood ratio orderlog-concavitynegative binomial mixturePoisson mixturestochastic comparisonstochastic orders

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60E05: Distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 244 - 254
Copyright
Copyright © Applied Probability Trust 2009 

References

Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 1019–1022.Google Scholar
Alamatsaz, M. H. and Abbasi, S. (2008). Ordering comparison of negative binomial random variables with their mixtures. Statist. Prob. Lett. 78, 2234–2239.CrossRefGoogle Scholar
Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Schur properties of convolutions of exponential and geometric random variables. J. Multivariate Anal. 48, 157–167.Google Scholar
Boland, P. J., Singh, H. and Cukic, B. (2002). Stochastic orders in partition and random testing of software. J. Appl. Prob. 39, 555–565.Google Scholar
Bon, J.-L. and Păltănea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5, 185–192.Google Scholar
Hardy, G. H., Littlewood, J. E. and Polya, G. (1964). Inequalities. Cambridge University Press.Google Scholar
Khaledi, B.-E. and Kochar, S. C. (2004). Ordering convolutions of gamma random variables. Sankhyā 66, 466–473.Google Scholar
Kochar, S. C. and Ma, C. (1999). Dispersive ordering of convolutions of exponential random variables. Statist. Prob. Lett. 43, 321–324. (Erratum: 45 (1999), 283.)CrossRefGoogle Scholar
Korwar, R. M. (2002). On stochastic orders for sums of independent random variables. J. Multivariate Anal. 80, 344–357.Google Scholar
Letac, G., Massam, H. and Richards, D. (2001). An expectation formula for the multivariate Dirichlet distribution. J. Multivariate Anal. 77, 117–137.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Mauldon, J. G. (1959). A generalization of the beta-distribution. Ann. Math. Statist. 30, 509–520.CrossRefGoogle Scholar
Misra, N., Singh, H. and Harner, E. J. (2003). Stochastic comparisons of poisson and binomial random variables with their mixtures. Statist. Prob. Lett. 65, 279–290.CrossRefGoogle Scholar
Shaked, M. (1980). On mixtures from exponential families. J. R. Statist. Soc. B 42, 192–198.Google Scholar
Shaked, M. and Shantikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Shaked, M. and Shantikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. J. Appl. Prob. 22, 619–633.CrossRefGoogle Scholar
Yu, Y. (2008). Relative log-concavity and a pair of triangle inequalities. {Tech. Rep.} Department of Statistics, University of California, Irvine.Google Scholar