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A note on the mixture representation of the conditional residual lifetime of a coherent system. (English) Zbl 1301.62023
Summary: This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

MSC:
 62E15 Exact distribution theory in statistics 60K10 Applications of renewal theory (reliability, demand theory, etc.)
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References:
 [1] Balakrishnan, N., Belzunce, F., Hami, N. and Khaledi, B. E. (2010). Univariate and multivariate likelihood ratio ordering of generalized order statistics and associated conditional variables. Prob. Eng. Inf. Sci. 24 , 441-455. · Zbl 1210.62050 [2] Belzunce, F., Gurler, S. and Ruiz, J. M. (2011). Revisiting multivariate likelihood ratio ordering results for order statistics. Prob. Eng. Inf. Sci. 25 , 355-368. · Zbl 1231.60015 [3] Ery\ilmaz, S. (2013). On residual lifetime of coherent systems after the $$r$$th failure. Statist. Papers 54 , 243-250. · Zbl 1256.62057 [4] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2012). Two-dimensional signatures. J. Appl. Prob. 49 , 416-429. · Zbl 1242.90070 [5] Goliforushani, S., Asadi, M. and Balakrishnan, N. (2012). On the residual and inactivity times of the components of used coherent systems. J. Appl. Prob. 49 , 385-404. · Zbl 1244.90074 [6] Karlin, S. (1968). Total Positivity , Vol. I. Stanford University Press. · Zbl 0219.47030 [7] Khaledi, B.-E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137 , 1173-1184. · Zbl 1111.60012 [8] Kochar, S. and Xu, M. (2010). On residual lifetimes of $$k$$-out-of-$$n$$ systems with nonidentical components. Prob. Eng. Inf. Sci. 24 , 109-127. · Zbl 1190.90060 [9] 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. · Zbl 0948.90067 [10] Li, X. and Zhang, Z. (2008). Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Business Industry 24 , 541-549. · Zbl 1198.62147 [11] Mahmoudi, M. and Asadi, M. (2011). The dynamic signature of coherent systems. IEEE Trans. Reliab. 60 , 817-822. [12] Misra, N. and van der Meulen, E. C. (2003). On stochastic properties of $$m$$-spacings. J. Statist. Planning Infer. 115 , 683-697. · Zbl 1016.62060 [13] Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45 , 1097-1112. · Zbl 1155.60305 [14] Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2010). The joint signature of coherent systems with shared components. J. Appl. Prob. 47 , 235-253. · Zbl 1187.60012 [15] Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2011). Signature-based representations for the reliability of systems with heterogeneous components. J. Appl. Prob. 48 , 856-867. · Zbl 1250.62052 [16] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R -34, 69-72. · Zbl 0585.62169 [17] Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York. · Zbl 1154.62075 [18] Samaniego, F. J., Balakrishnan, N. and Navarro, J. (2009). Dynamic signatures and their use in comparing the reliability of new and used systems. Naval Res. Logistics 56 , 577-591. · Zbl 1182.90036 [19] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders . Springer, New York. · Zbl 0806.62009 [20] Zhang, Z. (2010). Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Prob. 47 , 876-885. · Zbl 1196.62130 [21] Zhang, Z. (2011). Ordering new conditional residual lifetimes of $$k$$-out-of-$$n$$ systems. Commun. Statist. Theory Meth. 40 , 1591-1600. · Zbl 1220.62127 [22] Zhang, Z. and Li, X. (2010). Some new result on stochastic orders and aging properties of coherent systems. IEEE Trans. Reliab. 59 , 718-724.
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