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Applications of the hazard rate ordering in reliability and order statistics

Part of: Survival analysis and censored data Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Philip J. Boland ,
Emad El-Neweihi  and
Frank Proschan
Philip J. Boland*
Affiliation:
University College, Dublin
Emad El-Neweihi*
Affiliation:
University of Illinois at Chicago
Frank Proschan*
Affiliation:
The Florida State University
*
∗ Postal address: Department of Statistics, University College Dublin. Belfield, Dublin 4, Ireland.
∗∗ Postal address: Department of Mathematics, University of Illinois at Chicago, IL 60680, USA.
∗∗∗ Postal address: Department of Statistics, The Florida State University, Tallahassee, FL 323063033, USA.
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Abstract

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T1, · ··, Tn) to the vector (T′1, · ··, T′n), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ1r(t), λ2r(t))), the more diverse (λ1, λ2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering.

The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n is the lifetime of a k-out-of-n system, then τ k|n is greater than τ k+1|n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, · ··, Tn, we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk +1:n is greater than Tk:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Keywords

STOCHASTIC ORDERHAZARD RATE ORDERLIKELIHOOD RATIO ORDERCOHERENT SYSTEMSk-OUT-OF-n SYSTEMSPROPORTIONAL HAZARD RATES

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 180 - 192
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by Air Force Office of Scientific Research under Grant No. AFOSR 91–0048.

Research supported in part by AFOSR Grant No. 91–0048.

Research supported in part by AFOSR Grant No. 91–0048.

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