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Bounds on Variances of Lifetimes of Coherent and Mixed Systems

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

Published online by Cambridge University Press:  14 July 2016

Krzysztof Jasiński ,
Jorge Navarro  and
Tomasz Rychlik
Krzysztof Jasiński*
Affiliation:
Nicolaus Copernicus University
Jorge Navarro*
Affiliation:
Universidad de Murcia
Tomasz Rychlik*
Affiliation:
Nicolaus Copernicus University and Polish Academy of Sciences
*
∗Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12, 87100 Toruń, Poland. Email address: krzys@mat.uni.torun.pl
∗∗Postal address: Facultad de Matematicas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
∗∗∗Postal address: Institute of Mathematics, Polish Academy of Sciences, Chopina 12, 87100 Toruń, Poland. Email address: trychlik@impan.gov.pl
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Abstract

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We consider coherent and mixed reliability systems composed of elements with independent and identically distributed lifetimes. We present upper bounds on variances of system lifetimes, expressed in terms of variances of single components. We also discuss attainability conditions and some special cases and examples.

Keywords

Coherent systemmixed systemvarianceSamaniego signatureupper bound

MSC classification

Secondary: 62N05: Reliability and life testing 60E15: Inequalities; stochastic orderings 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 894 - 908
Copyright
Copyright © Applied Probability Trust 2009 

References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Beare, B. K. (2009). A generalization of Hoeffding's lemma, and a new class of covariance inequalities. Statist. Prob. Lett. 79, 637–642.Google Scholar
Block, H. W. and Fang, Z. B. (1988). A multivariate extension of Hoeffding's lemma. Ann. Prob. 16, 1803–1820.Google Scholar
Boland, P. J. and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective (Internat. Ser. Operat. Res. Manag. Sci. 67), eds Soyer, R., Mazzuchi, T. and Singpurwalla, N., Kluwer, Boston, MA, pp. 3–30.Google Scholar
Cuadras, C. M. (2002). On the covariance between functions. J. Multivariate Anal. 81, 19–27.Google Scholar
Hoeffding, W. (1940). Masstabinvariante korrelations-theorie. Schriften Math. Inst. Univ. Berlin 5, 181–233.Google Scholar
Jones, M. C. and Balakrishnan, N. (2002). How are moments and moments of spacings related to distribution functions? J. Statist. Planning Infer. 103, 377–390.CrossRefGoogle Scholar
Klimczak, M. and Rychlik, T. (2004). Maximum variance of kth records. Statist. Prob. Lett. 69, 421–430.Google Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507–523.3.0.CO;2-D>CrossRefGoogle Scholar
Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.Google Scholar
Mardia, K. V. and Thompson, J. W. (1972). Unified treatment of moment-formulae. Sankhyā 34, 121–132.Google Scholar
Navarro, J. and Rubio, R. (2009). Computations of signatures of coherent systems with five components. To appear in Commun. Statist. Simul. Comput. Google Scholar
Navarro, J. and Rychlik, T. (2007). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.CrossRefGoogle Scholar
Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008a). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.CrossRefGoogle Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008b). On the application and extension of system signatures to problems in engineering reliability. Naval Res. Logistics 55, 313–327.Google Scholar
Papadatos, N. (1995). Maximum variance of order statistics. Ann. Inst. Statist. Math. 47, 185–193.CrossRefGoogle Scholar
Papadatos, N. (1997). A note on maximum variance of order statistics from symmetric populations. Ann. Inst. Statist. Math. 49, 117–121.Google Scholar
Prakasa Rao, B. L. S. (1998). Hoeffding identity, multivariance and multicorrelation. Statistics 32, 13–29.Google Scholar
Quesada-Molina, J. J. (1992). A generalization of an identity of Hoeffding and some applications. J. Italian Statist. Soc. 3, 405–411.Google Scholar
Rychlik, T. (2001). Projecting Statistical Functionals (Lecture Notes Statist. 160). Springer, New York.CrossRefGoogle Scholar
Rychlik, T. (2008a). Bounds on lifetimes of coherent systems with exchangeable components. In Advances in Mathematical Modeling for Reliability, eds Bedford, T. et al., IOS Press, Amsterdam, pp. 111–118.Google Scholar
Rychlik, T. (2008b). Extreme variances of order statistics in dependent samples. Statist. Prob. Lett. 78, 1577–1582.CrossRefGoogle Scholar
Samaniego, F. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 69–72.CrossRefGoogle Scholar
Samaniego, F. (2007). System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci. 110). Springer, New York.CrossRefGoogle Scholar
Schoenberg, I. J. (1959). On variation diminishing approximation methods. In On Numerical Approximation (Proc. Symp., Madison, 1958), ed. Langer, R. E., University of Wisconsin Press, Madison, pp. 249–274.Google Scholar
Shaked, M. and Suarez-Llorens, A. (2003). On the comparison of reliability experiments based on the convolution order. J. Amer. Statist. Assoc. 98, 693–702.Google Scholar
Yu, H. (1993). A Glivenko–Cantelli lemma and weak convergence for empirical processes of associated sequences. Prob. Theory Relat. Fields 95, 357–370.Google Scholar