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Preservation of multivariate stochastic orders under multivariate Poisson shock models

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Tityik Wong
Tityik Wong*
Affiliation:
Community College of Southern Nevada
*
∗Postal address: Department of Mathematics, Community College of Southern Nevada, 3200 East Cheyenne Avenue-51A, North Las Vegas, NV 89030, USA.
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Abstract

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i(t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γm are independent of Mk = (Mk,1, · ··, Mk,m), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M1 and M2 are transformed into some stochastic orders between Z1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γm are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.

Keywords

MULTIVARIATE SHOCK MODELSPRESERVATION PROPERTIESUPPER-ORTHANT ORDERLOWER-ORTHANT ORDERUPPER-ORTHANT CONVEX ORDERLOWER-ORTHANT CONCAVE ORDERUSUAL STOCHASTIC ORDERINCREASING CONVEX ORDERINCREASING CONCAVE ORDERLIKELIHOOD RATIO ORDERLAPLACE TRANSFORM ORDER; INCREASING COMPONENT-WISE CONVEX ORDERCOMPONENT-WISE CONVEX ORDERMOMENTS ORDERNON-HOMOGENEOUS POISSON PROCESS

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.) 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 1009 - 1020
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Supported by NSF Grant DMS 9303891.

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