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On unimodality of the lifetime distribution of coherent systems with failure-dependent component lifetimes

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

Published online by Cambridge University Press:  26 July 2018

M. Bieniek  and
M. Burkschat
M. Bieniek*
Affiliation:
Maria Curie Skłodowska University
M. Burkschat*
Affiliation:
RWTH Aachen University
*
* Postal address: Institute of Mathematics, Maria Curie Skłodowska University, Pl. Marii Curie Skłodowskiej 1, 20-031 Lublin, Poland. Email address: mariusz.bieniek@umcs.lublin.pl
** Postal address: Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany. Email address: marco.burkschat@rwth-aachen.de
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Abstract

We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some examples for different types of systems. In particular, coherent systems with strictly bimodal density functions are presented in the case of independent standard uniform distributed lifetimes of components. Furthermore, we use the results to derive a sharp upper bound on the expected system lifetime in terms of the mean and the standard deviation of the underlying distribution.

Keywords

Unimodalitybimodalitysignaturecoherent systemsequential order statisticsgeneralized order statisticsvariation diminishing propertybounds

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60E05: Distributions 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 473 - 487
Copyright
Copyright © Applied Probability Trust 2018 

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References

Aki, S. and Hirano, K. (1997). Lifetime distributions of consecutive-k-out-of-n:F systems. Nonlinear Anal. 30, 555–562. Google Scholar
Alam, K. (1972). Unimodality of the distribution of an order statistic. Ann. Math. Statist. 43, 2041–2044. Google Scholar
Alimohammadi, M. and Alamatsaz, M. H. (2011). Some new results on unimodality of generalized order statistics and their spacings. Statist. Prob. Lett. 81, 1677–1682. Google Scholar
Alimohammadi, M., Alamatsaz, M. H. and Cramer, E. (2016). Convolutions and generalization of logconcavity: implications and applications. Naval Res. Logistics 63, 109–123. Google Scholar
Balakrishnan, N., Beutner, E. and Kamps, U. (2011). Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans. Reliab. 60, 605–611. Google Scholar
Barlow, R. E. and Proschan, F. (1966). Inequalities for linear combinations of order statistics from restricted families. Ann. Math. Statist. 37, 1574–1592. Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. Google Scholar
Bieniek, M. (2007). Variation diminishing property of densities of uniform generalized order statistics. Metrika 65, 297–309. Google Scholar
Bieniek, M. (2008). On families of distributions for which optimal bounds on expectations of GOS can be derived. Commun. Statist. Theory Meth. 37, 1997–2009. Google Scholar
Bieniek, M. (2016). Comparison of the bias of trimmed and Winsorized means. Commun. Statist. Theory Meth. 45, 6641–6650. Google Scholar
Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603. Google Scholar
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl. Prob. 46, 1052–1072. Google Scholar
Burkschat, M. and Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287. Google Scholar
Burkschat, M. and Navarro, J. (2018). Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. To appear in Prob. Eng. Inf. Sci. Google Scholar
Chen, H., Xie, H. and Hu, T. (2009). Log-concavity of generalized order statistics. Statist. Prob. Lett. 79, 396–399. Google Scholar
Cramer, E. (2003). Contributions to generalized order statistics. Habilitation Thesis, University of Oldenburg. Google Scholar
Cramer, E. (2004). Logconcavity and unimodality of progressively censored order statistics. Statist. Prob. Lett. 68, 83–90. Google Scholar
Cramer, E. (2016). Sequential order statistics. Wiley StatsRef: Statistics Reference Online, 7pp. Google Scholar
Cramer, E. and Kamps, U. (2001). Sequential k-out-of-n systems. In Advances in Reliability (Handbook Statist. 20), North-Holland, Amsterdam, pp. 301–372. Google Scholar
Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310. Google Scholar
Cramer, E., Kamps, U. and Rychlik, T. (2002a). Evaluations of expected generalized order statistics in various scale units. Appl. Math. (Warsaw) 29, 285–295. Google Scholar
Cramer, E., Kamps, U. and Rychlik, T. (2002b). On the existence of moments of generalized order statistics. Statist. Prob. Lett. 59, 397–404. Google Scholar
Cramer, E., Kamps, U. and Rychlik, T. (2004). Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann. Inst. Statist. Math. 56, 183–192. Google Scholar
D'Andrea, A. and De Sanctis, L. (2015). The Kruskal-Katona theorem and a characterization of system signatures. J. Appl. Prob. 52, 508–518. Google Scholar
Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA. Google Scholar
Dziubdziela, W. and Kopociński, B. (1976). Limiting properties of the k-th record values. Zastos. Mat. 15, 187–190. Google Scholar
Esary, J. D. and Marshall, A. W. (1970). Coherent life functions. SIAM J. Appl. Math. 18, 810–814. Google Scholar
Gajek, L. and Rychlik, T. (1996). Projection method for moment bounds on order statistics from restricted families. I. Dependent case. J. Multivariate Anal. 57, 156–174. Google Scholar
Hollander, M. and Peña, E. A. (1995). Dynamic reliability models with conditional proportional hazards. Lifetime Data Anal. 1, 377–401. Google Scholar
Huang, J. S. and Ghosh, M. (1982). A note on strong unimodality of order statistics. J. Amer. Statist. Assoc. 77, 929–930. (Correction: 78 (1983), 1008.) Google Scholar
Jasiński, K., Navarro, J. and Rychlik, T. (2009). Bounds on variances of lifetimes of coherent and mixed systems. J. Appl. Prob. 46, 894–908. Google Scholar
Kamps, U. (1995). A concept of generalized order statistics. J. Statist. Planning Inference 48, 1–23. Google Scholar
Kamps, U. (2016). Generalized order statistics. Wiley StatsRef: Statistics Reference Online, 12pp. Google Scholar
Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280. Google Scholar
Marichal, J.-L., Mathonet, P. and Waldhauser, T. (2011). On signature-based expressions of system reliability. J. Multivariate Anal. 102, 1410–1416. Google Scholar
Moriguti, S. (1953). A modification of Schwarz's inequality, with applications to distributions. Ann. Math. Statist. 24, 107–113. Google Scholar
Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135. Google Scholar
Navarro, J. and Rubio, R. (2010). Computations of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84. Google Scholar
Ross, S. M., Shahshahani, M. and Weiss, G. (1980). On the number of component failures in systems whose component lives are exchangeable. Math. Operat. Res. 5, 358–365. Google Scholar
Rychlik, T. (2001). Projecting Statistical Functionals (Lecture Notes Statist. 160). Springer, New York. Google Scholar
Sabnis, S. V. and Nair, M. R. (1997). Coherent structures and unimodality. J. Appl. Prob. 34, 812–817. Google Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72. Google Scholar
Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York. Google Scholar