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coherent system; order statistic; copula; exchangeable distribution; absolute continuous distribution; absolute continuous copula
Performance of coherent reliability systems is strongly connected with distributions of order statistics of failure times of components. A crucial assumption here is that the distributions of possibly mutually dependent lifetimes of components are exchangeable and jointly absolutely continuous. Assuming absolute continuity of marginals, we focus on properties of respective copulas and characterize the marginal distribution functions of order statistics that may correspond to absolute continuous and possibly exchangeable copulas. One characterization is based on the vector of distribution functions of all order statistics, and the other concerns the distribution of a single order statistic.
[1] Barlow R. E., Proschan F.: Mathematical Theory of Reliability. Wiley, New York 1965 MR 0195566 | Zbl 0874.62111
[2] Barlow R. E., Proschan F.: Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart and Winston, New York 1975 MR 0438625 | Zbl 0379.62080
[3] Bassan B., Spizzichino F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93 (2005), 313–339 DOI 10.1016/j.jmva.2004.04.002 | MR 2162641 | Zbl 1070.60015
[4] Boland P. J., Samaniego F. J.: The signature of a coherent system and its applications in reliability. In: Mathematical Reliability Theory: An Expository Perspective (R. Soyer, T. Mazzuchi, and N. Singpurwalla, eds.), Kluwer Academic Publishers, Boston 2004, pp. 1–29 MR 2064996
[5] David H. A., Nagaraja H. N.: Order Statistics. Third edition. Wiley, Hoboken, NJ 2003 MR 1994955 | Zbl 1053.62060
[6] Durante F., Jaworski P.: Absolutely continuous copulas with given diagonal sections. Comm. Statist. Theory Methods 37 (2008), 18, 2924–2942 DOI 10.1080/03610920802050927 | MR 2467742
[7] Durante F., Kolesárová A., Mesiar, R., Sempi C.: Copulas with given diagonal sections: novel constructions and applications. Internat. J. Uncertainty, Fuzziness, and Knowledge-Based Systems 15 (2007), 397–410 DOI 10.1142/S0218488507004753 | MR 2362234 | Zbl 1158.62324
[8] Durante F., Mesiar, R., Sempi C.: On a family of copulas constructed from the diagonal section. Soft Comput. 10 (2006), 490–494 DOI 10.1007/s00500-005-0523-7 | Zbl 1098.60016
[9] Galambos J.: The role of exchangeability in the theory of order statistics. In: Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.), North-Holland, Amsterdam 1982, pp. 75–87 MR 0675966 | Zbl 0505.62027
[10] Genest C., Quesada-Molina J. J., Rodríguez-Lallena J. A., Sempi C.: A characterization of quasi-copulas. J. Multivariate Anal. 69 (1999), 193–205 DOI 10.1006/jmva.1998.1809 | MR 1703371 | Zbl 0935.62059
[11] Jaworski P.: On uniform tail expansions of multivariate copulas and wide convergence of measures. Appl. Math. 33 (2006), 159–184 MR 2267746 | Zbl 1102.62053
[12] Jaworski P.: On copulas and their diagonals. Inform. Sci., to appear MR 2547755 | Zbl 1171.62332
[13] Joe H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London 1997 MR 1462613 | Zbl 0990.62517
[14] Kochar S., Mukerjee, H., Samaniego F. J.: The “signature” of a coherent system and its application to comparisons among systems. Naval Res. Logist. 46 (1999), 507–523 DOI 10.1002/(SICI)1520-6750(199908)46:5<507::AID-NAV4>3.0.CO;2-D | MR 1700160 | Zbl 0948.90067
[15] Lai C. D., Xie M.: Stochastic Ageing and Dependence for Reliability. Springer-Verlag, New York 2006 MR 2223811 | Zbl 1098.62130
[16] Mesiar R., Sempi C.: Ordinal sums and idempotents of copulas. Aequationaes Math., to appear MR 2640277 | Zbl 1205.62063
[17] Navarro J., Rychlik T.: Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98 (2007), 102–113 DOI 10.1016/j.jmva.2005.09.003 | MR 2292919 | Zbl 1102.62111
[18] Navarro J., Samaniego F. J., Balakrishnan, N., Bhattacharya D.: On the application and extension of system signatures in engineering reliability. Nav. Res. Logist. 55 (2008), 314–326 DOI 10.1002/nav.20285 | MR 2402184 | Zbl 1153.90386
[19] Nelsen R. B.: An Introduction to Copulas. (Lecture Notes in Statistics 139.) Springer, New York 1999 DOI 10.1007/978-1-4757-3076-0_1 | MR 1653203 | Zbl 1152.62030
[20] Rychlik T.: Bounds for expectation of $L$-estimates for dependent samples. Statistics 24 (1993), 1–7 DOI 10.1080/02331888308802385 | MR 1238259 | Zbl 0808.62048
[21] Rychlik T.: Distributions and expectations of order statistics for possibly dependent random variables. J. Multivariate Anal. 48 (1994), 31–42 DOI 10.1016/0047-259X(94)80003-E | MR 1256833 | Zbl 0790.62048
[22] Samaniego F. J.: On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34 (1985), 69–72 DOI 10.1109/TR.1985.5221935 | Zbl 0585.62169
[23] Sklar A.: Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229–231 MR 0125600
[24] Spizzichino F.: Subjective probability models for lifetimes. (Monographs on Statistics and Applied Probability 91.) Chapman and Hall/CRC, Boca Raton 2001 MR 1980207 | Zbl 1078.62530
[25] Spizzichino F.: The role of symmetrization and signature for systems with non-exchangeable components. In: Advances in Mathematical Modelling for Reliability (T. Bedford, J. Quigley, L. Walls, B. Alkali, A. Daneshkhah, and G. Hardman, eds.), IOS Press, Amsterdam 2008, pp. 138–148 MR 2464240
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