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STOCHASTIC COMPARISONS OF SYSTEMS BASED ON SEQUENTIAL ORDER STATISTICS VIA PROPERTIES OF DISTORTED DISTRIBUTIONS

Published online by Cambridge University Press:  27 February 2017

M. Burkschat  and
J. Navarro
M. Burkschat
Affiliation:
Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany E-mail: marco.burkschat@rwth-aachen.de
J. Navarro
Affiliation:
Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain E-mail: jorgenav@um.es
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Abstract

We consider systems based on sequential order statistics (SOS) with underlying distributions possessing proportional hazard rates (PHRs). In that case, the lifetime distribution of the system can be expressed as a distorted distribution. Motivated by the distribution structure in the case of pairwise different model parameters, a particular class of distorted distributions, the generalized PHR model, is introduced and characterizations of stochastic comparisons for several stochastic orders are obtained. Moreover, results on the asymptotic behavior of some aging characteristics, for example, the hazard rate and the mean residual life function, of general distorted distributions as well as related bounds are given. The results are supplemented with limiting properties of the systems in the case of possibly equal model parameters. Some examples are presented in order to illustrate the application of the findings to systems based on SOS and also to systems with independent heterogeneous components.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 2 , April 2018 , pp. 246 - 274
Copyright
Copyright © Cambridge University Press 2017 

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References

1. Aki, S. & Hirano, K. (1997). Lifetime distributions of consecutive-k-out-of-n: F systems. Nonlinear Analysis 30: 555–562.CrossRefGoogle Scholar
2. Arnold, B.C., Balakrishnan, N., & Nagaraja, H.N. (1998). Records. New York: Wiley.CrossRefGoogle Scholar
3. Balakrishnan, N., Beutner, E., & Kamps, U. (2011). Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Transactions on Reliability TR-60: 605–611.CrossRefGoogle Scholar
4. Balakrishnan, N. & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27: 403–443.CrossRefGoogle Scholar
5. Barlow, R.E. & Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With. Maryland: Silver Spring.Google Scholar
6. Block, H., Dugas, M., & Samaniego, F.J. (2007). Signature-related results on system lifetimes. In Nair, V. (ed.), Advances in Statistical Modeling and Inference: Essays in Honor of Kjell A. Doksum. Singapore: World Scientific, pp. 115–129.CrossRefGoogle Scholar
7. Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. Journal of Applied Probability 46: 1052–1072.CrossRefGoogle Scholar
8. Burkschat, M. & Lenz, B. (2009). Marginal distributions of the counting process associated with generalized order statistics. Communications in Statistics – Theory and Methods 38: 2089–2106.CrossRefGoogle Scholar
9. Burkschat, M. & Navarro, J. (2011). Aging properties of sequential order statistics. Probability in the Engineering and Informational Sciences 25: 449–467.CrossRefGoogle Scholar
10. Burkschat, M. & Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. Journal of Applied Probability 50: 272–287.CrossRefGoogle Scholar
11. Burkschat, M. & Navarro, J. (2014). Asymptotic behavior of the hazard rate in systems based on sequential order statistics. Metrika 77: 965–994.CrossRefGoogle Scholar
12. Burkschat, M. & Torrado, N. (2014). On the reversed hazard rate of sequential order statistics. Statistics and Probability Letters 85: 106–113.CrossRefGoogle Scholar
13. Cramer, E. (2004). Logconcavity and unimodality of progressively censored order statistics. Journal of Statistical Planning and Inference 68: 83–90.Google Scholar
14. Cramer, E. (2006). Sequential order statistics. In Kotz, S., Balakrishnan, N., Read, C.B., Vidakovic, B., & Johnson, N.L. (eds.), Encyclopedia of Statistical Sciences, vol. 12, 2nd ed. Hoboken: Wiley, pp. 7629–7634.Google Scholar
15. Cramer, E. & Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Annals of the Institute of Statistical Mathematics 48: 535–549.CrossRefGoogle Scholar
16. Cramer, E. & Kamps, U. (2001). Sequential k-out-of-n systems. In Balakrishnan, N. & Rao, C.R. (eds.), Handbook of statistics – advances in reliability, vol. 20. Amsterdam: Elsevier, pp. 301–372.Google Scholar
17. Cramer, E. & Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310.CrossRefGoogle Scholar
18. Cramer, E., Kamps, U., & Rychlik, T. (2004). Unimodality of uniform generalized order statistics, with applications to mean bounds. Annals of the Institute of Statistical Mathematics 56: 183–192.CrossRefGoogle Scholar
19. Dziubdziela, W. & Kopociński, B. (1976). Limiting properties of the k-th record values. Applicationes Mathematicae 15: 187–190.CrossRefGoogle Scholar
20. Esary, J.D. & Marshall, A.W. (1970). Coherent life functions. SIAM Journal on Applied Mathematics 18: 810–814.CrossRefGoogle Scholar
21. Hall, W.J. & Wellner, J.A. (1981). Mean residual life. In Csörgö, M., Dawson, D.A., Rao, J.N.K., & Saleh, A.K.Md.E. (eds.), Statistics and related topics. Amsterdam: North-Holland Publishing Company.Google Scholar
22. Hollander, M. & Peña, E.A. (1995). Dynamic reliability models with conditional proportional hazard rates. Lifetime Data Analysis 1: 377–401.CrossRefGoogle Scholar
23. Hürlimann, W. (2004). Distortion risk measures and economic capital. North American Actuarial Journal 8: 86–95.CrossRefGoogle Scholar
24. Kamps, U. (1995). A concept of generalized order statistics. Journal of Statistical Planning and Inference 48: 1–23.CrossRefGoogle Scholar
25. Kamps, U. & Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35: 269–280.CrossRefGoogle Scholar
26. Kozyra, P.M. & Rychlik, T. (2016). Sharp bounds on the expectations of L-statistics expressed in the Gini mean difference units. To appear in Communications in Statistics – Theory and Methods Published online first (Apr 2016), 46: 2921–2941. DOI: 10.1080/03610926.2015.1053937.CrossRefGoogle Scholar
27. Liu, Q., Mao, T., & Hu, T. (2014). The second-order regular variation of order statistics. Probability in the Engineering and Informational Sciences 28: 209–222.CrossRefGoogle Scholar
28. Luke, Y.L. (1969). The Special Functions and Their Approximations, vol. I. New York: Academic Press.Google Scholar
29. Mathai, A.M. (1993). A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford: Clarendon Press.Google Scholar
30. Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. TEST 25: 150–169.CrossRefGoogle Scholar
31. Navarro, J. & Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Research Logistics 58: 123–135.CrossRefGoogle Scholar
32. Navarro, J., del Aguila, Y., Sordo, M.A., & Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry 29: 264–278.CrossRefGoogle Scholar
33. Navarro, J., del Aguila, Y., Sordo, M.A., & Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Applied Stochastic Models in Business and Industry 30: 444–454.CrossRefGoogle Scholar
34. Navarro, J. & Gomis, M.C. (2016). Comparisons in the mean residual life order of coherent systems with identically distributed components. Applied Stochastic Models in Business and Industry 32: 33–47.CrossRefGoogle Scholar
35. Navarro, J. & Rubio, R. (2009). Computations of signatures of coherent systems with five components. Communications in Statistics – Simulation and Computation 39: 68–84.CrossRefGoogle Scholar
36. Navarro, J., Samaniego, F.J., Balakrishnan, N., & Bhattacharya, D. (2008). On the application and extension of system signatures in engineering reliability. Naval Research Logistics 55: 313–327.CrossRefGoogle Scholar
37. Samaniego, F.J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability TR-34: 69–72.CrossRefGoogle Scholar
38. Samaniego, F.J. (2007). System Signatures and their Applications in Engineering Reliability. New York: Springer.CrossRefGoogle Scholar
39. Torrado, N., Lillo, R.E., & Wiper, M.P. (2012). Sequential order statistics: ageing and stochastic orderings. Methodology and Computing in Applied Probability 14: 579–596.CrossRefGoogle Scholar