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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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