Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 25, 2020

Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

  • Giovanna Nappo EMAIL logo and Fabio Spizzichino
From the journal Dependence Modeling


We first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing.

A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework.

In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

MSC 2010: 60K10; 60E15; 62E10; 62H05; 60G09; 91B30


[1] Arjas, E. (1981). A stochastic process approach to multivariate reliability systems: notions based on conditional stochastic order. Math. Oper. Res. 6(2), 263–276.10.1287/moor.6.2.263Search in Google Scholar

[2] Arjas, E. and I. Norros (1984). Life lengths and association: a dynamic approach. Math. Oper. Res. 9(1), 151–158.10.1287/moor.9.1.151Search in Google Scholar

[3] Arjas, E. and I. Norros (1991). Stochastic order and martingale dynamics in multivariate life length models: a review. In K. Mosler and M. Scarsini (Eds.), Stochastic Orders and Decision under Risk, pp. 7–24. Inst. Math. Statist., Hayward CA.10.1214/lnms/1215459846Search in Google Scholar

[4] Avérous, J. and J.-L. Dortet-Bernadet (2004). Dependence for Archimedean copulas and aging properties of their generating functions. Sankhya A 66(4), 607–620.Search in Google Scholar

[5] Barlow, R. E. (1985). A Bayes explanation of an apparent failure rate paradox. IEEE Trans. Reliab. 34(2), 107–108.10.1109/TR.1985.5221964Search in Google Scholar

[6] Barlow, R. E. and M. B. Mendel (1992). De Finetti-type representations for life distributions. J. Amer. Statist. Assoc. 87(420), 1116–1122.10.1080/01621459.1992.10476267Search in Google Scholar

[7] Barlow, R. E. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Search in Google Scholar

[8] Barlow, R. E. and F. Spizzichino (1993). Schur-concave survival functions and survival analysis. J. Comput. Appl. Math. 46(3), 437–447.10.1016/0377-0427(93)90039-ESearch in Google Scholar

[9] Bassan, B. and F. Spizzichino (1999). Stochastic comparisons for residual lifetimes and Bayesian notions of multivariate ageing. Adv. in Appl. Probab. 31(4), 1078–1094.10.1239/aap/1029955261Search in Google Scholar

[10] Bassan, B. and F. Spizzichino (2001). Dependence and multivariate aging: the role of level sets of the survival function. In Y. Hayakawa, T. Irony and M. Xie (Eds.), System and Bayesian Reliability, pp. 229–242. World Sci. Publ., River Edge NJ.10.1142/9789812799548_0013Search in Google Scholar

[11] Bassan, B. and F. Spizzichino (2003). On some properties of dependence and aging for residual lifetimes in the exchangeable case. In B. H. Lindqvist and K. A. Doksum (Eds.), Mathematical and Statistical Methods in Reliability, pp. 235–249. World Sci. Publ., River Edge NJ.10.1142/9789812795250_0016Search in Google Scholar

[12] Bassan, B. and F. Spizzichino (2005). Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93(2), 313–339.10.1016/j.jmva.2004.04.002Search in Google Scholar

[13] Birnbaum, Z. W., J. D. Esary, and A. W. Marshall (1966). A stochastic characterization of wear-out for components and systems. Ann. Math. Statist. 37, 816–825.10.1214/aoms/1177699362Search in Google Scholar

[14] Capéraà, P., A.-L. Fougères, and C. Genest (1997). A stochastic ordering based on a decomposition of Kendall’s tau. In V. Beneš and J. Št¥pàn (Eds.), Distributions with Given Marginals and Moment Problems, pp. 81–86. Kluwer Acad. Publ., Dordrecht.10.1007/978-94-011-5532-8_9Search in Google Scholar

[15] Capéraà, P., A.-L. Fougères, and C. Genest (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72(1), 30–49.10.1006/jmva.1999.1845Search in Google Scholar

[16] Cooray, K. (2018). Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family. Depend. Model. 6, 1–18.10.1515/demo-2018-0001Search in Google Scholar

[17] de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7(1), 1–68.Search in Google Scholar

[18] Durante, F., R. Foschi, and F. Spizzichino (2010). Aging functions and multivariate notions of NBU and IFR. Probab. Engrg. Inform. Sci. 24(2), 263–278.10.1017/S026996480999026XSearch in Google Scholar

[19] Durante, F. and R. Ghiselli-Ricci (2009). Supermigrative semi-copulas and triangular norms. Inform. Sci. 179(15), 2689–2694.10.1016/j.ins.2009.04.001Search in Google Scholar

[20] Durante, F., A. Kolesárová, R. Mesiar, and C. Sempi (2007). Copulas with given diagonal sections: novel constructions and applications. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 15(4), 397–410.10.1142/S0218488507004753Search in Google Scholar

[21] Durante, F. and C. Sempi (2005a). Copula and semicopula transforms. Int. J. Math. Math. Sci. (4), 645–655.10.1155/IJMMS.2005.645Search in Google Scholar

[22] Durante, F. and C. Sempi (2005b). Semicopulæ. Kybernetika 41(3), 315–328.Search in Google Scholar

[23] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL.Search in Google Scholar

[24] Durante, F. and F. Spizzichino (2010). Semi-copulas, capacities and families of level sets. Fuzzy Sets Syst. 161(2), 269–276.10.1016/j.fss.2009.03.002Search in Google Scholar

[25] Foschi, R. and F. Spizzichino (2012). Interactions between ageing and risk properties in the analysis of burn-in problems. Decis. Anal. 9(2), 103–118.10.1287/deca.1120.0236Search in Google Scholar

[26] Foschi, R. and F. Spizzichino (2013). Reversing conditional orderings. In I. Haijun and I. Xiaohu (Eds.), Stochastic Orders in Reliability and Risk, pp. 59–80. Springer New York.10.1007/978-1-4614-6892-9_3Search in Google Scholar

[27] Genest, C. and L.-P. Rivest (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88(423), 1034–1043.10.1080/01621459.1993.10476372Search in Google Scholar

[28] Genest, C. and L.-P. Rivest (2001). On the multivariate probability integral transformation. Statist. Probab. Lett. 53(4), 391–399.10.1016/S0167-7152(01)00047-5Search in Google Scholar

[29] Hürlimann, W. (2004). Properties and measures of dependence for the archimax copula. Adv. Appl. Stat. 5(2), 125–143.Search in Google Scholar

[30] Janson, S., T. Konstantopoulos, and L. Yuan (2016). On a representation theorem for finitely exchangeable random vectors. J. Math. Anal. Appl. 442(2), 703–714.10.1016/j.jmaa.2016.04.070Search in Google Scholar

[31] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Search in Google Scholar

[32] Kerns, G. J. and G. J. Székely (2006). De Finetti’s theorem for abstract finite exchangeable sequences. J. Theoret. Probab. 19(3), 589–608.10.1007/s10959-006-0028-zSearch in Google Scholar

[33] Kimeldorf, G. and A. R. Sampson (1989). A framework for positive dependence. Ann. Inst. Statist. Math. 41(1), 31–45.Search in Google Scholar

[34] Lai, C.-D. and M. Xie (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Search in Google Scholar

[35] Leonetti, P. (2016). Finite partially exchangeable laws are signed mixtures of product laws. Sankhya A 80(2), 195–214.10.1007/s13171-017-0123-5Search in Google Scholar

[36] Li, H. and X. Li (2013). Stochastic Orders in Reliability and Risk Management. In Honor of Professor Moshe Shaked. Springer, New York.Search in Google Scholar

[37] Littlewood, B. (1984). Subjective probability and the dfr-mixture closure theorem. Comm. Statist. Theory Methods 13(7), 859–863.10.1080/03610928408828724Search in Google Scholar

[38] Mai, J.-F. (2019). The infinite extendibility problem for exchangeable real-valued random vectors. Available at in Google Scholar

[39] Marshall, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York-London.Search in Google Scholar

[40] Müller, A. and M. Scarsini (2005). Archimedean copulae and positive dependence. J. Multivariate Anal. 93(2), 434–445.10.1016/j.jmva.2004.04.003Search in Google Scholar

[41] Nappo, G. and F. Spizzichino (2009). Kendall distributions and level sets in bivariate exchangeable survival models. Inform. Sci. 179(17), 2878–2890.10.1016/j.ins.2009.02.007Search in Google Scholar

[42] Navarro, J., J. M. Ruiz, and Y. del Aguila (2008). Characterizations and ordering properties based on log-odds functions. Statistics 42(4), 313–328.10.1080/02331880701835762Search in Google Scholar

[43] Navarro, J. and M. A. Sordo (2018). Stochastic comparisons and bounds for conditional distributions by using copula properties. Depend. Model. 6, 156–177.10.1515/demo-2018-0010Search in Google Scholar

[44] Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Search in Google Scholar

[45] Nelsen, R. B., J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and M. Úbeda-Flores (2003). Kendall distribution functions. Statist. Probab. Lett. 65(3), 263–268.10.1016/j.spl.2003.08.002Search in Google Scholar

[46] Nelsen, R. B., J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and M. Úbeda Flores (2009). Kendall distribution functions and associative copulas. Fuzzy Sets Syst. 160(1), 52–57.10.1016/j.fss.2008.05.001Search in Google Scholar

[47] Scarsini, M. and M. Shaked (1996). Positive dependence orders: a survey. In C.C. Heyde, Y. V. Prohorov, R. Pyke, S.T. Rachev (Eds.), Athens Conference on Applied Probability and Time Series Analysis, pp. 70–91. Springer, New York.10.1007/978-1-4612-0749-8_5Search in Google Scholar

[48] Scarsini, M. and F. Spizzichino (1999). Simpson-type paradoxes, dependence, and ageing. J. Appl. Probab. 36(1), 119–131.10.1017/S0021900200016892Search in Google Scholar

[49] Shaked, M. and J. G. Shanthikumar (2007). Stochastic Orders. Springer, New York.10.1007/978-0-387-34675-5Search in Google Scholar

[50] Spizzichino, F. (1992). Reliability decision problems under conditions of ageing. In J. M. Bernardo, J. O. Berger, A. P. Dawid, A. F. M. Smith (Eds.), Bayesian Statistics 4, pp. 803–811. Oxford Univ. Press.Search in Google Scholar

[51] Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman & Hall/CRC, Boca Raton FL.10.1201/9781420036138Search in Google Scholar

[52] Spizzichino, F. (2010). Semi-copulas and interpretations of coincidences between stochastic dependence and ageing. In P. Jaworski, F. Durante, W. Härdle W., T. Rychlik (Eds.), Copula Theory and its Applications, pp. 237–254. Springer, Heidelberg.10.1007/978-3-642-12465-5_11Search in Google Scholar

[53] Spizzichino, F. (2014). Aging and Positive Dependence. Wiley StatsRef: Statistics Reference Online. Available at in Google Scholar

[54] Spreeuw, J. (2014). Archimedean copulas derived from utility functions. Insurance Math. Econom. 59, 235–242.10.1016/j.insmatheco.2014.10.002Search in Google Scholar

Received: 2019-05-24
Accepted: 2020-01-21
Published Online: 2020-03-25

© 2020 Giovanna Nappo et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 4.2.2023 from
Scroll Up Arrow