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Multivariate monotone likelihood ratio and uniform conditional stochastic order

Published online by Cambridge University Press:  14 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Bell Laboratories
*
∗Postal address: Bell Laboratories, WB-1A350, Bell Laboratories, Holmdel, NJ 07733, U.S.A.
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Abstract

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Keywords

STOCHASTIC ORDERCONDITIONAL PROBABILITYUNIFORM CONDITIONAL STOCHASTIC ORDERTOTAL POSITIVITYMONOTONE LIKELIHOOD RATIOSUPERMODULARITYFKG INEQUALITY
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 695 - 701
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Arjas, E. (1981) A stochastic process approach to multivariate reliability systems: notions based on conditional stochastic order. Math. Operat. Res. 6, 263–276.CrossRefGoogle Scholar
[2] Fahmy, S., De Pereira, B. C. A., Proschan, F. and Shared, M. (1982) The influence of the sample on the posterior distribution. Commun. Statist. To appear.Google Scholar
[3] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.Google Scholar
[4] Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities: I. Multivariate totally positive distributions. J. Multivariate Analysis 10, 467–498.Google Scholar
[5] Keilson, J. and Sumita, U. (1983) Uniform stochastic ordering and related inequalities. Canad. J. Statist. To appear.Google Scholar
[6] Kemperman, J. H. B. (1977) On the FKG inequality for measures on a partially ordered space. Indag. Math. 39, 313–331.Google Scholar
[7] Milgrom, P. R. (1981) Good news and bad news: representation theorems and applications. Bell J. Econom. 12, 380–391.Google Scholar
[8] Milgrom, P. R. and Weber, R. J. (1982) A theory of auctions and competitive bidding. Econometrica. To appear.Google Scholar
[9] Topkis, D. M. (1968) Ordered Optimal Solutions. Ph.D. dissertation, Department of Operations Research, Stanford University.Google Scholar
[10] Topkis, D. M. (1978) Minimizing a submodular function on a lattice. Operat. Res. 26, 305–321.Google Scholar
[11] Whitt, W. (1979) A note on the influence of the sample on the posterior distribution. J. Amer. Statist. Assoc. 74, 424–426.Google Scholar
[12] Whitt, W. (1980) Uniform conditional stochastic order. J. Appl. Prob. 17, 112–123.Google Scholar
[13] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.Google Scholar
[14] Whitt, W. (1981a) The renewal-process stationary-excess operator. Bell Laboratories.Google Scholar