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Univariate and multivariate stochastic comparisons and ageing properties of the generalized Pólya process

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  28 March 2018

F. G. Badía ,
C. Sangüesa  and
Ji Hwan Cha
F. G. Badía*
Affiliation:
University of Zaragoza
C. Sangüesa*
Affiliation:
University of Zaragoza
Ji Hwan Cha*
Affiliation:
Ewha Womans University
*
* Postal address: Department of Statistical Methods, University of Zaragoza, Zaragoza, 50009, Spain.
* Postal address: Department of Statistical Methods, University of Zaragoza, Zaragoza, 50009, Spain.
**** Postal address: Department of Statistics, Ewha Womans University, Seoul, 120-750, Korea. Email address: jhcha@ewha.ac.kr
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Abstract

In this work we consider the generalized Pólya process with baseline intensity function r and parameters α and β recently studied by Cha (2014). The aim of this work is to provide both univariate and multivariate stochastic comparisons between two generalized Pólya processes with different baseline intensity functions and the same parameters α and β for the epoch and inter-epoch times of the two processes. The comparisons are analogous to stochastic comparisons in Belzunce et al. (2001) for two nonhomogeneous Poisson or pure-birth processes with different intensity functions. Moreover, we study both univariate and multivariate ageing properties of the epoch and inter-epoch times of the generalized Pólya process.

Keywords

Generalized Pólya processepoch timestochastic orderageing propertygeneralized order statistics

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 233 - 253
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Arias-Nicolás, J. P., Fernández-Ponce, J. M., Luque-Calvo, P. and Suárez-Llorens, A. (2005). Multivariate dispersion order and the notion of copula applied to the multivariate t-distribution. Prob. Eng. Inf. Sci. 19, 363–375. CrossRefGoogle Scholar
[2]Asfaw, Z. G. and Lindqvist, B. H. (2015). Extending minimal repair models for repairable systems: a comparison of dynamic and heterogeneous extensions of a nonhomogeneous Poisson process. Reliab. Eng. System Safety 140, 53–58. CrossRefGoogle Scholar
[3]Babykina, G. and Couallier, V. (2010). Modelling recurrent events for repairable systems under worse than old assumption. In Advances in Degradation Modeling, Birkhäuser, Boston, MA, pp. 339–354. CrossRefGoogle Scholar
[4]Badía, F. G. (2011). Hazard rate properties of a general counting process stopped at an independent random time. J. Appl. Prob. 48, 56–67. CrossRefGoogle Scholar
[5]Badía, F. G. and Sangüesa, C. (2016). Negative ageing properties for counting processes arising in virtual age models. Appl. Math. Modelling 40, 5271–5282. CrossRefGoogle Scholar
[6]Balakrishnan, N., Belzunce, F., Sordo, M. A. and Suárez-Llorens, A. (2012). Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J. Multivariate Anal. 105, 45–54. CrossRefGoogle Scholar
[7]Bassan, B. and Spizzichino, F. (1999). Stochastic comparisons for residual lifetimes and Bayesian notions of multivariate ageing. Adv. Appl. Prob. 31, 1078–1094. CrossRefGoogle Scholar
[8]Baxter, L. A. (1982). Reliability applications of relevation transform. Naval. Res. Logistics Quart. 29, 321–330. CrossRefGoogle Scholar
[9]Belzunce, F. and Ruiz, J.-M. (2002). Multivariate dispersive ordering of epoch times of nonhomogeneous Poisson processes. J. Appl. Prob. 39, 637–643. CrossRefGoogle Scholar
[10]Belzunce, F. and Shaked, M. (2001). Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory. Statist. Prob. Lett. 53, 363–372. CrossRefGoogle Scholar
[11]Belzunce, F., Hu, T. and Khaledi, B.-E. (2003). Dispersion-type variability orders. Prob. Eng. Inf. Sci. 17, 305–334. CrossRefGoogle Scholar
[12]Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016). An Introduction to Stochastic Orders. Elsevier, Amsterdam. Google Scholar
[13]Belzunce, F., Mercader, J. A. and Ruiz, J. M. (2003). Multivariate aging properties of epoch times of nonhomogeneous processes. J. Multivariate Anal. 84, 335–350. CrossRefGoogle Scholar
[14]Belzunce, F., Mercader, J.-A. and Ruiz, J.-M. (2005). Stochastic comparisons of generalized order statistics. Prob. Eng. Inf. Sci. 19, 99–120. CrossRefGoogle Scholar
[15]Belzunce, F., Suárez-Llorens, A. and Sordo, M. A. (2012). Comparison of increasing directionally convex transformations of random vectors with a common copula. Insurance Math. Econom. 50, 385–390. CrossRefGoogle Scholar
[16]Belzunce, F., Lillo, R. E., Ruiz, J. M. and Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Prob. Eng. Inf. Sci. 15, 199–224. CrossRefGoogle Scholar
[17]Burkschat, M. (2009). Multivariate dependence of spacings of generalized order statistics. J. Multivariate Anal. 100, 1093–1106. CrossRefGoogle Scholar
[18]Cha, J. H. (2014). Characterization of the generalized Pólya process and its applications. Adv. Appl. Prob. 46, 1148–1171. CrossRefGoogle Scholar
[19]Cha, J. H. and Finkelstein, M. (2016). New shock models based on generalized Pólya process. Europ. J. Operat. Res. 251, 135–141. CrossRefGoogle Scholar
[20]Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310. CrossRefGoogle Scholar
[21]Di Crescenzo, A. and Spina, S. (2016). Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process. Math. Biosci. 282, 121–134. CrossRefGoogle Scholar
[22]Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York. Google Scholar
[23]Finkelstein, M. and Esaulova, V. (2005). On the weak IFR aging of bivariate lifetime distributions. Appl. Stoch. Models Business Industry 21, 265–272. CrossRefGoogle Scholar
[24]Franco, M., Ruiz, J. M. and Ruiz, M. C. (2002). Stochastic orderings between spacings of generalized order statistics. Prob. Eng. Inf. Sci. 16, 471–484. CrossRefGoogle Scholar
[25]Kamps, U. (1995). A concept of generalized order statistics. J. Statist. Planning Infer. 48, 1–23. CrossRefGoogle Scholar
[26]Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press. Google Scholar
[27]Khaledi, B.-E. and Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems. J. Appl. Prob. 37, 1123–1128. CrossRefGoogle Scholar
[28]Khaledi, B.-E. and Shaked, M. (2010). Stochastic comparisons of multivariate mixtures. J. Multivariate Anal. 101, 2486–2498. CrossRefGoogle Scholar
[29]Khaledi, B.-E., Farsinezhad, S. and Kochar, S. C. (2011). Stochastic comparisons of order statistics in the scale model. J. Statist. Planning Infer. 141, 276–286. CrossRefGoogle Scholar
[30]Kochar, S. C. (1996). Some results on interarrival times of nonhomogeneous Poisson processes. Prob. Eng. Inf. Sci. 10, 75–85. CrossRefGoogle Scholar
[31]Konno, H. (2010). On the exact solution of a generalized Pólya process. Adv. Math. Phys. 2010, 504267. CrossRefGoogle Scholar
[32]Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York. Google Scholar
[33]Le Gat, Y. (2009). Une extension du processus de Yule pour la modélisation stochastique des événements récurrents. Application aux défaillances de canalisations d'eau sous pression. Doctoral thesis, Agro Paris Tech. Google Scholar
[34]Le Gat, Y. (2014). Extending the Yule process to model recurrent pipe failures in water supply networks. Urban Water J. 11, 617–630. CrossRefGoogle Scholar
[35]Lee, H. and Cha, J. H. (2016). New stochastic models for preventive maintenance and maintenance optimization. Europ. J. Operat. Res. 255, 80–90. CrossRefGoogle Scholar
[36]Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York. Google Scholar
[37]Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester. Google Scholar
[38]Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. TEST 25, 150–169. CrossRefGoogle Scholar
[39]Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113. CrossRefGoogle Scholar
[40]Navarro, J. and Shaked, M. (2010). Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika 71, 313–317. CrossRefGoogle Scholar
[41]Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Europ. J. Operat. Res. 240, 127–139. Google Scholar
[42]Pellerey, F., Shaked, M. and Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Prob. Eng. Inf. Sci. 14, 353–373. CrossRefGoogle Scholar
[43]Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. 34, 335–343. Google Scholar
[44]Rinott, Y. and Samuel-Cahn, E. (1992). Optimal stopping values and prophet inequalities for some dependent random variables. In Stochastic Inequalities (IMS Lecture Notes Monogr. Ser. 22), Institute of Mathematical Statistics, Hayward, CA, pp. 343–358. CrossRefGoogle Scholar
[45]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. CrossRefGoogle Scholar
[46]Tan, W. Y. (1986). A stochastic Gompertz birth-death process. Statist. Prob. Lett. 4, 25–28. CrossRefGoogle Scholar
[47]Wang, Y. and Zhao, P. (2010). A note on DRHR preservation property of generalized order statistics. Commun. Statist. Theory Meth. 39, 815–822. CrossRefGoogle Scholar
[48]Xie, H. and Hu, T. (2009). Ordering p-spacings of generalized order statistics revisited. Prob. Eng. Inf. Sci. 23, 1–16. CrossRefGoogle Scholar