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Extreme Dispersions of Semicoherent and Mixed System Lifetimes

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

Published online by Cambridge University Press:  30 January 2018

Patryk Miziuła  and
Tomasz Rychlik
Patryk Miziuła*
Affiliation:
Nicolaus Copernicus University
Tomasz Rychlik*
Affiliation:
Polish Academy of Sciences
*
∗ Postal address: Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Chopina 12, 87-100 Torun, Poland.
∗∗ Postal address: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland. Email address: [email protected]
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Abstract

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We consider semicoherent and mixed systems with exchangeable components. We present sharp lower and upper bounds on various dispersion measures (in particular, variance and median absolute deviation) of the system lifetime, expressed in terms of the system signature and the dispersion of a single component lifetime. We construct joint exchangeable distributions of component lifetimes with two-point marginals which attain the bounds in the limit.

Keywords

Semicoherent systemmixed systemSamaniego signatureexchangeable random variablessharp bounddispersion measurevariancemedian absolute deviation

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60E15: Inequalities; stochastic orderings 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 117 - 128
Copyright
© Applied Probability Trust 

References

Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.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), Kluwer, Boston, MA, pp. 3–30.Google Scholar
Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems built with separate modules. J. Appl. Prob. 48, 843–855.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
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.Google Scholar
Miziuła, P. and Rychlik, T. (2014). Sharp bounds for lifetime variances of reliability systems with exchangeable components. IEEE Trans. Reliab. 63, 850–857.Google Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures 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.Google Scholar
Rychlik, T. (1993). Bounds for expectation of L-estimates for dependent samples. Statistics 24, 1–7.CrossRefGoogle Scholar
Rychlik, T. (1994). Distributions and expectations of order statistics for possibly dependent random variables. J. Multivariate Anal. 48, 31–42.Google Scholar
Rychlik, T. (2008). Extreme variances of order statistics in dependent samples. Statist. Prob. Lett. 78, 1577–1582.Google Scholar
Rychlik, T. (2012). Applications of Samaniego signatures to bounds on variances of coherent and mixed system lifetimes. In Recent Advances in System Reliability, Springer, London, pp. 63–77.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 (Internat. Ser. Operat. Res. Manag. Sci. 110). Springer, New York.Google Scholar
Uhlmann, W. (1963). Ranggrössen als Schätzfunktionen. Metrika 7, 23–40.CrossRefGoogle Scholar
Zieliński, R. (1999). Best equivariant nonparametric estimator of a quantile. Statist. Prob. Lett. 45, 79–84.Google Scholar
Zieliński, R. (2001). PMC-optimal nonparametric quanitle esitmator. Statistics 35, 453–462.Google Scholar