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Competing risks and shock models governed by a generalized bivariate Poisson process
Published online by Cambridge University Press: 15 December 2022
Abstract
We propose a stochastic model for the failure times of items subject to two external random shocks occurring as events in an underlying bivariate counting process. This is a special formulation of the competing risks model, which is of interest in reliability theory and survival analysis. Specifically, we assume that a system, or an item, fails when the sum of the two types of shock reaches a critical random threshold. In detail, the two kinds of shock occur according to a bivariate space-fractional Poisson process, which is a two-dimensional vector of independent homogeneous Poisson processes time-changed by an independent stable subordinator. Various results are given, such as analytic hazard rates, failure densities, the probability that the failure occurs due to a specific type of shock, and the survival function. Some special cases and ageing notions related to the NBU characterization are also considered. In this way we generalize certain results in the literature, which can be recovered when the underlying process reduces to the homogeneous Poisson process.
MSC classification
Primary:
60E05: Distributions
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- Original Article
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust
References
Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Applied Mathematics Series). Dover Publications.Google Scholar
Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Bedford, T. and Cooke, R. (2001). Probabilistic Risk Analysis: Foundations and Methods. Cambridge University Press.CrossRefGoogle Scholar
Beghin, L. and Macci, C. (2016). Multivariate fractional Poisson processes and compound sums. Adv. Appl. Prob. 48, 691–711.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2016). New shock models based on the generalized Polya process. Europ. J. Operat. Res. 251, 135–141.CrossRefGoogle Scholar
Cha, J. H. and Giorgio, M. (2018). Modelling of marginally regular bivariate counting process and its application to shock model. Methodology Comput. Appl. Prob. 20, 1137–1154.CrossRefGoogle Scholar
Crowder, M. J. (2001). Classical Competing Risks. Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Di Crescenzo, A. and Longobardi, M. (2006). On the NBU ageing notion within the competing risks model. J. Statist. Planning Infer. 136, 1638–1654.CrossRefGoogle Scholar
Di Crescenzo, A. and Longobardi, M. (2008). Competing risks within shock models. Sci. Math. Jpn. 67, 125–135.Google Scholar
Di Crescenzo, A. and Meoli, A. (2017). Competing risks driven by Mittag–Leffler distributions, under copula and time transformed exponential model. Ric. Mat. 66, 361–381.CrossRefGoogle Scholar
Di Crescenzo, A. and Pellerey, F. (2019). Some results and applications of geometric counting processes. Methodology Comput. Appl. Prob. 21, 203–233.CrossRefGoogle Scholar
Garra, R., Orsingher, E. and Scavino, M. (2017). Some probabilistic properties of fractional point processes. Stoch. Anal. Appl. 35, 701–718.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Elsevier / Academic Press, Amsterdam.Google Scholar
Guo, B.-N. and Qi, F. (2011). The function
$(b^{x}-a^{x})/x$
: logarithmic convexity and applications to extended mean values. Filomat 25, 63–73.CrossRefGoogle Scholar
$(b^{x}-a^{x})/x$
: logarithmic convexity and applications to extended mean values. Filomat 25, 63–73.CrossRefGoogle ScholarJohnson, W. P. (2002). The curious history of Faà di Bruno’s formula. Amer. Math. Monthly 109, 217–234.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies 204). Elsevier Science, Amsterdam.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families (Springer Series in Statistics). Springer, New York.Google Scholar
Michelitsch, T. M. and Riascos, A. P. (2020). Generalized fractional Poisson process and related stochastic dynamics. Fract. Calc. Appl. Anal. 23, 656–693.CrossRefGoogle Scholar
Navarro, J. (2022). Introduction to System Reliability Theory. Springer Nature Switzerland AG, Cham.CrossRefGoogle Scholar
Orsingher, E. and Polito, F. (2012). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852–858.CrossRefGoogle Scholar
Orsingher, E. and Toaldo, B. (2015). Counting processes with Bernštein intertimes and random jumps. J. Appl. Prob. 52, 1028–1044.CrossRefGoogle Scholar
Polito, F. and Scalas, E. (2016). A generalization of the space-fractional Poisson process and its connection to some Lévy processes. Electron Commun. Prob. 21, 20.CrossRefGoogle Scholar
Uchaikin, V. and Zolotarev, V. M. (1999). Chance and Stability: Stable Distributions and their Applications. VSP, Utrecht.CrossRefGoogle Scholar