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Transformed Lévy processes as state-dependent wear models

Part of: Special processes Distribution theory - Probability Stochastic 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: jhcha@ewha.ac.kr
**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: sophie.mercier@univ-pau.fr
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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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References

Abdel-Hameed, M. (1975). A gamma wear process. IEEE Trans. Reliab . 24, 152–153.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016). An Introduction to Stochastic Orders. Academic Press, Amsterdam.Google Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1982). Negative Dependence (Lecture Notes Monogr. Ser. 2). Institute of Mathematical Statistics, Hayward, CA, pp. 206–215.Google Scholar
Cha, J. H. (2014). Characterization of the generalized Pólya process and its applications. Adv. Appl. Prob. 46, 1148–1171.CrossRefGoogle Scholar
Chhikara, R. S. and Folks, J. L. (1977). The inverse gaussian distribution as a lifetime model. Technometrics 19, 461–468.CrossRefGoogle Scholar
žinlar, E., Bažant, Z. P. and Osman, E. (1977). Stochastic process for extrapolating concrete creep. J. Eng. Mech. Div. 103, 1069–1088.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2006). Actuarial Theory for Dependent Risks: Masures, Orders and Models. John Wiley, Chichester.Google Scholar
Doksum, K. A. and Hóyland, A. (1992). Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution. Technometrics 34, 74–82.CrossRefGoogle Scholar
Giorgio, M. and Pulcini, G. (2018). A new state-dependent degradation process and related model misidentification problems. Europ. J. Operat. Res. 267, 1027–1038.CrossRefGoogle Scholar
Giorgio, M., Guida, M. and Pulcini, G. (2010). A state-dependent wear model with an application to marine engine cylinder liners. Technometrics 52, 172–187.CrossRefGoogle Scholar
Giorgio, M., Guida, M. and Pulcini, G. (2015). A new class of Markovian processes for deteriorating units with state dependent increments and covariates. IEEE Trans. Reliab . 64, 562–578.CrossRefGoogle Scholar
Giorgio, M., Guida, M. and Pulcini, G. (2018). The transformed gamma process for degradation phenomena in presence of unexplained forms of unit-to-unit variability. Quality Reliab. Eng. Internat. 34, 543–562.CrossRefGoogle Scholar
Kahle, W., Mercier, S. and Paroissin, C. (2016). Degradation Processes in Reliability. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. II. Multivariate reverse rule distributions. J. Multivariate Anal. 10, 499–516.CrossRefGoogle Scholar
Lemoine, A. J. and Wenocur, M. L. (1985). On failure modeling. Naval Res. Logistics Quart . 32, 497–508.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Science and Business Media, New York.Google Scholar
Mohtashami Borzadaran, G. R. and Mohtashami Borzadaran, H. A. (2011). Log-concavity property for some well-known distributions. Surveys Math. Appl . 6, 203–219.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Rinott, Y. and Samuel-Cahn, E. (1992). Optimal stopping values and prophet inequalities for some dependent random variables. (Lecture Notes Monogr. Ser. 22). Institute of Mathematical Statistics, Hayward, CA, pp. 343–358.Google Scholar
Ross, S. M. (2006). Introduction to Probability Models, 9th edn. Academic Press.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions Camb. Stud. Adv. Math. 68. Cambridge University Press.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Singpurwalla, N. (1995). Survival in dynamic environments. Statist. Sci. 10, 86–103.CrossRefGoogle Scholar
Van Noortwijk, J. M. (2009). A survey of the application of gamma processes in maintenance. Reliab. Eng. System Safety 94, 2–21.CrossRefGoogle Scholar
Wang, X. and Xu, D. (2010). An inverse gaussian process model for degradation data. Technometrics 52, 188–197.CrossRefGoogle Scholar
Wenocur, M. L. (1989). A reliability model based on the gamma process and its analytic theory. Adv. Appl. Prob. 21, 899–918.CrossRefGoogle Scholar
Whitmore, G. A. (1995). Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime Data Anal . 1, 307–319.CrossRefGoogle ScholarPubMed
Ye, Z.-S. and Chen, N. (2014). The inverse gaussian process as a degradation model. Technometrics 56, 302–311.CrossRefGoogle Scholar