Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-03T19:13:55.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic ordering of classical discrete distributions

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Achim Klenke  and
Lutz Mattner
Achim Klenke*
Affiliation:
Johannes Gutenberg-Universität Mainz
Lutz Mattner*
Affiliation:
Universität Trier
*
∗ Postal address: Johannes Gutenberg-Universität Mainz, Institut für Mathematik, Staudingerweg 9, 55099 Mainz, Germany. Email address: [email protected]
∗∗ Postal address: Universität Trier, FB IV - Mathematik, 54286 Trier, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k*})/Q({k*}) ≥ 1 ≥ limk→k*P({k})/Q({k}), with k* and k* denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k*})/Q({k*}) for finite k*. This includes in particular all pairs where P and Q are both binomial (bn1,p1 ≤stbn2,p2 if and only if n1 ≤ n2 and (1 - p1)n1 ≥ (1 - p2)n2, or p1 = 0), both negative binomial (b−r1,p1 ≤stb−r2,p2 if and only if p1 ≥ p2 and p1r1 ≥ p2r2), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

Keywords

Bernoulli convolutionbinomial distributioncouplinghypergeometric distributionnegative binomial distributionmonotone likelihood ratiooccupancy problemPascal distributionPoisson distributionstochastic orderingwaiting times

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 392 - 410
Copyright
Copyright © Applied Probability Trust 2010 

References

Boland, P. J., Singh, H. and Cukic, B. (2002). Stochastic orders in partition and random testing of software. J. Appl. Prob. 39, 555–565.CrossRefGoogle Scholar
Gastwirth., J. L. (1977). A probability model of a pyramid scheme. Amer. Statist. 31, 79–82.Google Scholar
Klenke, A. (2008). Probability Theory. Springer, London.CrossRefGoogle Scholar
Ma, C. (1997). A note on stochastic ordering of order statistics. J. Appl. Prob. 34, 785–789.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Pfanzagl, J. (1964). On the topological structure of some ordered families of distributions. Ann. Math. Statist. 35, 1216–1228.CrossRefGoogle Scholar
Proschan, F. and Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616.CrossRefGoogle Scholar
Rüschendorf, L. (1991). On conditional stochastic ordering of distributions. Adv. Appl. Prob. 23, 46–63.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1993). Stochastic monotonicity and Slepian-type inequalities for infinitely divisible and stable random vectors. Ann. Prob. 21, 143–160.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.CrossRefGoogle Scholar
Vatutin, V. A. and Mikhajlov, V. G. (1982). Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory Prob. Appl. 27, 734–743.CrossRefGoogle Scholar
Whitt, W. (1980). Uniform conditional stochastic order. J. Appl. Prob. 17, 112–123.CrossRefGoogle Scholar
Wong, C. K. and Yue, P. C. (1973). A majorization theorem for the number of distinct outcomes in N independent trials. Discrete Math. 6, 391–398.CrossRefGoogle Scholar