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Transformed Lévy processes as state-dependent wear models

Part of: Stochastic processes Distribution theory - Probability Special processes

Published online by Cambridge University Press:  07 August 2019

Ji Hwan Cha  and
Sophie Mercier
Ji Hwan Cha*
Affiliation:
Ewha Womans University
Sophie Mercier*
Affiliation:
Université de Pau et des Pays de l’Adour
*
*Postal address: Department of Statistics, Ewha Womans University, Seoul, 120-750, Korea. Email address: [email protected]
**Postal address: Laboratoire de Mathématiques et de leurs Applications de Pau, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, Avenue de l’Université, BP 1155, 64013 Pau, France. Email address: [email protected]
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Abstract

Many wear processes used for modeling accumulative deterioration in a reliability context are nonhomogeneous Lévy processes and, hence, have independent increments, which may not be suitable in an application context. In this work we consider Lévy processes transformed by monotonous functions to overcome this restriction, and provide a new state-dependent wear model. These transformed Lévy processes are first observed to remain tractable Markov processes. Some distributional properties are derived. We investigate the impact of the current state on the future increment level and on the overall accumulated level from a stochastic monotonicity point of view. We also study positive dependence properties and stochastic monotonicity of increments.

Keywords

Reliabilitydeterioration modelwear processstochastic orderpositive dependencenonindependent increment

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60E15: Inequalities; stochastic orderings 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 468 - 486
Copyright
© Applied Probability Trust 2019 

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