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Failure rate properties of parallel systems

Part of: Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  15 July 2020

Idir Arab ,
Milto Hadjikyriakou  and
Paulo Eduardo Oliveira
Idir Arab*
Affiliation:
University of Coimbra, CMUC, Department of Mathematics, Portugal
Milto Hadjikyriakou*
Affiliation:
University of Central Lancashire, Cyprus
Paulo Eduardo Oliveira*
Affiliation:
University of Coimbra, CMUC, Department of Mathematics, Portugal
*
*Postal address: Department of Mathematics, PO Box 3008, EC Santa Cruz, Coimbra, Portugal.
**Postal address: School of Sciences, 12-14 University Avenue, Pyla, 7080Larnaka, Cyprus.
***Postal address: Department of Mathematics, PO Box 3008, EC Santa Cruz, Coimbra, Portugal. Email: paulo@mat.uc.pt
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Abstract

We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

Keywords

Convex stochastic orderiterated failure ratesign variation

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60E05: Distributions 62N05: Reliability and life testing
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 563 - 587
Copyright
© Applied Probability Trust 2020

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References

Abouammoh, A. and El-Neweihi, E. (1986). Closure of the NBUE and DMRL classes under formation of parallel systems. Statist. Prob. Lett. 4, 223–225.CrossRefGoogle Scholar
Abu-Youssef, S. E. (2002). A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application. Statist. Prob. Lett. 57, 171–177.CrossRefGoogle Scholar
Arab, I. and Oliveira, P. E. (2019). Iterated failure rate monotonicity and ordering relations within gamma and Weibull distributions. Prob. Eng. Inf. Sci. 33, 64–80.CrossRefGoogle Scholar
Arab, I. and Oliveira, P. E. (2018). Iterated failure rate monotonicity and ordering relations within gamma and Weibull distributions – corrigendum. Prob. Eng. Inf. Sci. 32, 640–641.CrossRefGoogle Scholar
Atkinson, A. B. (1970). On the measurement of inequality. J. Econom. Theory 2, 244–263.CrossRefGoogle Scholar
Averous, J. and Meste, M. (1989). Tailweight and life distributions. Statist. Prob. Lett. 8, 381–387.CrossRefGoogle Scholar
Balakrishnan, N., Haidari, A. and Masoumifard, K. (2015). Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Trans. Reliab. 64 (1), 333–348.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York, Montreal, London.Google Scholar
Bashkar, E., Torabi, H. and Asadi, M. (2017). Stochastic comparisons of series and parallel systems with heterogeneous components. Preprint. Available at http://arxiv.org/abs/1704.06329 .Google Scholar
Belzunce, F., Candel, J. and Ruiz, J. M. (1995). Ordering of truncated distributions through concentration curves. Sankhy A 57, 375–383.Google Scholar
Boutsikas, M. V. and Vaggelatou, E. (2002). On the distance between convex-ordered random variables, with applications. Adv. Appl. Prob. 34, 349–374.CrossRefGoogle Scholar
Bryson, M. C. and Siddiqui, M. M. (1969). Some criteria for aging. J. Amer. Statist. Assoc. 64, 1472–1483.CrossRefGoogle Scholar
Chatterjee, A. and Mukherjee, S. P. (2001). Equilibrium distribution – its role in reliability theory. In Handbook of Statistics, Vol. 20, eds Balakrishnan, N. and Rao, C.R., North-Holland, Amsterdam, pp. 105–137.CrossRefGoogle Scholar
Chen, Y. (1994). Classes of life distributions and renewal counting process. J. Appl. Prob. 31, 1110–1115.CrossRefGoogle Scholar
Cheng, K. and He, Z. F. (1989). Reliability bounds in NBUE and NWUE life distributions. Acta Math. Appl. Sin. 5, 81–88.CrossRefGoogle Scholar
Cheng, K. and Lam, Y. (2002). Reliability bounds on NBUE life distributions with known first two moments. Naval Res. Logistics 49, 781–797.CrossRefGoogle Scholar
Cox, D. R. (1962). Renewal Theory. Spottiswoode, Ballantyne and Co., London.Google Scholar
Deshpande, J. V., Kochar, S. C. and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–758.CrossRefGoogle Scholar
Dykstra, R., Kochar, S. and Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. J. Statist. Planning Infer. 65, 203–211.CrossRefGoogle Scholar
El-Bassiouny, A. H. (2003). On testing exponentiality against IFRA alternatives. Appl. Math. Comput. 146, 445–453.Google Scholar
Fagiuoli, E. and Pellerey, F. (1993). New partial orderings and applications. Naval Res. Logistics 40, 829–842.3.0.CO;2-D>CrossRefGoogle Scholar
Khaledi, B. E. and Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems. J. Appl. Prob. 37, 1123–1128.CrossRefGoogle Scholar
Kochar, S. C. and Wiens, D. D. (1987). Partial orderings of life distributions with respect to their ageing properties. Naval Res. Logistics 34, 823–829.3.0.CO;2-R>CrossRefGoogle Scholar
Kochar, S. C. and Xu, M. (2007a). Some recent results on stochastic comparisons and dependence among order statistics in the case of PHR model. J. Iranian Statist. Soc. 6, 125–140.Google Scholar
Kochar, S. C. and Xu, M. (2007b). Stochastic comparisons of parallel systems when components have proportional hazard rates. Prob. Eng. Inf. Sci. 21, 597–609.CrossRefGoogle Scholar
Kochar, S. C. and Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. J. Appl. Prob. 46, 342–352.CrossRefGoogle Scholar
Lando, T. and Bertoli-Barsotti, L. (2016). Weak orderings for intersecting Lorenz curves. Metron 74, 177–192.CrossRefGoogle Scholar
Loh, W. Y. (1984). Bounds on AREs for restricted classes of distributions defined via tail-orderings. Ann. Statist. 12, 685–701.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Application. Academic Press, New York.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Muliere, P. and Scarsini, M. (1989). A note on stochastic dominance and inequality measures. J. Econom. Theory 49, 314–323.CrossRefGoogle Scholar
Nanda, A. K., Hazra, N. K., Al-Mutairi, D. K. and Ghitany, M. E. (2017). On some generalized ageing orderings. Commun. Statist. Theory Meth. 46, 5273–5291.CrossRefGoogle Scholar
Navarro, J. and Hernandez, P. J. (2004). How to obtain bathtub-shaped failure rate models from normal mixtures. Prob. Eng. Inf. Sci. 18, 511–531.CrossRefGoogle Scholar
Ortobelli, S., Rachev, S., Shalit, H. and Fabozzi, F. J. (2008). Orderings and risk probability functionals in portfolio selection. Prob. Math. Statist. 28, 203–234.Google Scholar
Patel, J. K. (1983). Hazard rate and other classifications of distributions. In Encyclopedia in Statistical Sciences 3, John Wiley, New York, pp. 590–594.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1990). Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. Appl. Prob. 22, 350–374.CrossRefGoogle Scholar
Ross, S. M. (1983). Stochastic Processes. Wiley, New York.Google Scholar
Rausand, M. and Høyland, A. (2004). System Reliability Theory: Models, Statistical Methods, and Applications. Wiley, Hoboken, New Jersey.Google Scholar
Sengupta, D. (1994). Another look at the moment bounds on reliability. J. Appl. Prob. 31, 777–787.CrossRefGoogle Scholar
Shaked, S. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Singh, H. (1989). On partial orderings. Naval Res. Logistics 36, 103–110.3.0.CO;2-7>CrossRefGoogle Scholar
Singh, H. and Jain, K. (1989). Preservation of some partial orderings under Poisson shock models. Adv. Appl. Prob. 21, 713–716.CrossRefGoogle Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
Shestopaloff, Yu. K. (2011). Properties of sums of some elementary functions and their application to computational and modeling problems. Comput. Math. Math. Phys. 51, 699–712.CrossRefGoogle Scholar
Van Zwet, W. R. (1964). Convex transformations of random variables (Mathematical Centre Tracts 7). Mathematisch Centrum, Amsterdam.Google Scholar
Whitmore, G. A. (1970). Third degree stochastic dominance. Amer. Econom. Rev. 60, 457–459.Google Scholar