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Uniform conditional variability ordering of probability distributions

Published online by Cambridge University Press:  14 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, Crawfords Corner Road, Holmdel, NJ 07733, USA.
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Abstract

Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.

Keywords

DISPERSIONSPREADVARIABILITYSTOCHASTIC COMPARISONSSTOCHASTIC ORDERCONDITIONAL PROBABILITIESUNIFORM CONDITIONAL STOCHASTIC ORDERMONOTONE LIKELIHOOD RATIOLOG-CONCAVITYRELATIVE LOG-CONCAVITYNETWORKS OF QUEUESSTATIONARY-EXCESS DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 3 , September 1985 , pp. 619 - 633
Copyright
Copyright © Applied Probability Trust 1985 

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References

Anderson, T. W. and Samuels, S. M. (1965) Some inequalities among binomial and Poisson probabilities. Proc. 5th Berkeley Symp. Math. Statist. Prob. 1, 1–12.Google Scholar
Bickel, P. J. and Lehmann, E. L. (1976) Descriptive statistics for nonparametric models, III. Dispersion. Ann. Statist. 4, 1139–1158.Google Scholar
Bickel, P. J. and Lehmann, E. L. (1979) Descriptive statistics for nonparametric models. IV. Spread. Contributions to Statistics, Jaroslaw Hajek Memorial Volume, ed. Jureckova, J., Reidel, Dordrecht, 33–40.Google Scholar
Birnbaum, Z. W. (1948) On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.Google Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford, Ca.Google Scholar
Karlin, S. and Novikoff, A. (1963) Generalized convex inequalities. Pacific J. Math. 13, 1251–1279.Google Scholar
Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities: I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.Google Scholar
Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Keilson, J. and Sumita, U. (1982) Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–198.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Milgrom, P. R. (1981) Good news and bad news: representation theorems and applications. Bell. I. Econom. 12, 380–391.Google Scholar
Milgrom, P. R. and Weber, R. J. (1982) A theory of auctions and competitive building. Econometrica 50, 1089–1122.Google Scholar
Oja, H. (1981) On location, scale, skewness and kurtosis. Scand. J. Statist. 8, 154–168.Google Scholar
Sauer, C. H. and Chandy, K. M. (1981) Computer Systems Performance Modeling. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Shaked, M. (1980) On mixtures from exponential families. J. R. Statist. Soc. B 42, 192–198.Google Scholar
Shaked, M. (1982) Dispersive ordering of distributions. J. Appl. Prob. 19, 310–320.CrossRefGoogle Scholar
Shaked, M. (1985) Ordering distributions in dispersion. In Encyclopedia of Statistical Sciences 5, ed. Kotz, S. and Johnson, N. L. Wiley, New York.Google Scholar
Simons, G. (1980) Extension of the stochastic ordering property of likelihood ratios. Ann. Statist. 8, 833–839.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
Whitt, W. (1980) Uniform conditional stochastic order. J. Appl. Prob. 17, 112–123.CrossRefGoogle Scholar
Whitt, W. (1982) Multivariate monotone likelihood ratio and uniform conditional stochastic order. J. Appl. Prob. 19, 695–701.Google Scholar
Whitt, W. (1984) Open and closed models for networks of queues. AT&T Bell Lab. Tech. J. 63, 1911–1979.Google Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar
Yanagimoto, T. and Sibuya, M. (1976) Isotonic tests for spread and tail. Ann. Inst. Statist. Math. 28, 329–342.Google Scholar
Yanagimoto, T. and Sibuya, M. (1980) Comparisons of tails of distributions in models for estimating safe doses. Ann. Inst. Statist. Math. 32, 325–340.Google Scholar
Zahorjan, J. (1983) Workload representations in queueing models of computer systems. Proc. ACM Sigmetrics Conference, Minneapolis, August 1983, 70–81.Google Scholar