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Uniform conditional variability ordering of probability distributions

Uniform conditional variability ordering of probability distributions <jats:p>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 <jats:italic>f</jats:italic> on the real line is said to be less than or equal to another, g, in <jats:italic>uniform conditional variability order</jats:italic> (UCVO) if the ratio <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)<jats:italic>/g</jats:italic>(<jats:italic>x</jats:italic>) is unimodal with the model yielding a supremum, but <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)/<jats:italic>g</jats:italic>(<jats:italic>x</jats:italic>) is log-concave. This is illustrated in a comparison of open and closed queueing network models.</jats:p> http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Probability CrossRef

Uniform conditional variability ordering of probability distributions

Journal of Applied Probability , Volume 22 (3): 619-633 – Sep 1, 1985

Uniform conditional variability ordering of probability distributions


Abstract

<jats:p>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 <jats:italic>f</jats:italic> on the real line is said to be less than or equal to another, g, in <jats:italic>uniform conditional variability order</jats:italic> (UCVO) if the ratio <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)<jats:italic>/g</jats:italic>(<jats:italic>x</jats:italic>) is unimodal with the model yielding a supremum, but <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)/<jats:italic>g</jats:italic>(<jats:italic>x</jats:italic>) is log-concave. This is illustrated in a comparison of open and closed queueing network models.</jats:p>

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Publisher
CrossRef
ISSN
0021-9002
DOI
10.2307/3213865
Publisher site
See Article on Publisher Site

Abstract

<jats:p>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 <jats:italic>f</jats:italic> on the real line is said to be less than or equal to another, g, in <jats:italic>uniform conditional variability order</jats:italic> (UCVO) if the ratio <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)<jats:italic>/g</jats:italic>(<jats:italic>x</jats:italic>) is unimodal with the model yielding a supremum, but <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic> 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 <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>)/<jats:italic>g</jats:italic>(<jats:italic>x</jats:italic>) is log-concave. This is illustrated in a comparison of open and closed queueing network models.</jats:p>

Journal

Journal of Applied ProbabilityCrossRef

Published: Sep 1, 1985

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