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Resolving an old problem on the preservation of the IFR property under the formation of $k$-out-of-$n$ systems with discrete distributions

Part of: Survival analysis and censored data

Published online by Cambridge University Press:  16 October 2023

Mahdi Alimohammadi Open the ORCID record for Mahdi Alimohammadi [Opens in a new window]  and
Jorge Navarro Open the ORCID record for Jorge Navarro [Opens in a new window]
Mahdi Alimohammadi*
Affiliation:
Alzahra University
Jorge Navarro*
Affiliation:
Universidad de Murcia
*
*Postal address: Department of Statistics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran. Email address: m.alimohammadi@alzahra.ac.ir
**Postal address: Department of Statistics and Operation Research, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
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Abstract

More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k-out-of-n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\mathrm{e}}}^x)$. Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.

Keywords

Coherent systemsageing classesfailure ratelog-concavityorder statistics

MSC classification

Secondary: 62N05: Reliability and life testing
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 644 - 653
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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