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Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes

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  • Bassan, Bruno
  • Spizzichino, Fabio

Abstract

For a couple of lifetimes (X1,X2) with an exchangeable joint survival function , attention is focused on notions of bivariate aging that can be described in terms of properties of the level curves of . We analyze the relations existing among those notions of bivariate aging, univariate aging, and dependence. A goal and, at the same time, a method to this purpose is to define axiomatically a correspondence among those objects; in fact, we characterize notions of univariate and bivariate aging in terms of properties of dependence. Dependence between two lifetimes will be described in terms of their survival copula. The language of copulæ turns out to be generally useful for our purposes; in particular, we shall introduce the more general notion of semicopula. It will be seen that this is a natural object for our analysis. Our definitions and subsequent results will be illustrated by considering a few remarkable cases; in particular, we find some necessary or sufficient conditions for Schur-concavity of , or for IFR properties of the one-dimensional marginals. The case characterized by the condition that the survival copula of (X1,X2) is Archimedean will be considered in some detail. For most of our arguments, the extension to the case of n>2 is straightforward.

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  • Bassan, Bruno & Spizzichino, Fabio, 2005. "Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 313-339, April.
    • Handle: RePEc:eee:jmvana:v:93:y:2005:i:2:p:313-339
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    Cited by:

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    2. Jorge Navarro & Franco Pellerey & Julio Mulero, 2022. "On sums of dependent random lifetimes under the time-transformed exponential model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 879-900, December.
    3. Li, Xiaohu & Lin, Jianhua, 2011. "Stochastic orders in time transformed exponential models with applications," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 47-52, July.
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    5. Charpentier, Arthur & Segers, Johan, 2009. "Tails of multivariate Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1521-1537, August.
    6. Nappo Giovanna & Spizzichino Fabio, 2020. "Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property," Dependence Modeling, De Gruyter, vol. 8(1), pages 1-33, January.
    7. Zalzadeh, Saeed & Pellerey, Franco, 2016. "A positive dependence notion based on componentwise unimodality of copulas," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 51-57.
    8. Boczek, Michał & Hovana, Anton & Hutník, Ondrej & Kaluszka, Marek, 2021. "New monotone measure-based integrals inspired by scientific impact problem," European Journal of Operational Research, Elsevier, vol. 290(1), pages 346-357.
    9. Serkan Eryilmaz & Cihangir Kan & Fatih Akici, 2009. "Consecutive k‐within‐m‐out‐of‐n:F system with exchangeable components," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 503-510, September.
    10. Bassan, Bruno & Spizzichino, Fabio, 2005. "Bivariate survival models with Clayton aging functions," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 6-12, August.
    11. Nappo Giovanna & Spizzichino Fabio, 2020. "Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property," Dependence Modeling, De Gruyter, vol. 8(1), pages 1-33, January.
    12. Jorge Navarro & Julio Mulero, 2020. "Comparisons of coherent systems under the time-transformed exponential model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 255-281, March.
    13. Jorge Navarro & Maria Longobardi & Franco Pellerey, 2017. "Comparison results for inactivity times of k-out-of-n and general coherent systems with dependent components," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 822-846, December.
    14. Genest Christian & Scherer Matthias, 2020. "The gentleman copulist: An interview with Carlo Sempi," Dependence Modeling, De Gruyter, vol. 8(1), pages 34-44, January.
    15. Mulero, Julio & Pellerey, Franco & Rodríguez-Griñolo, Rosario, 2010. "Stochastic comparisons for time transformed exponential models," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 328-333, April.
    16. Li, Chen & Li, Xiaohu, 2015. "Likelihood ratio order of sample minimum from heterogeneous Weibull random variables," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 46-53.
    17. Longobardi, Maria & Pellerey, Franco, 2019. "On the role of dependence in residual lifetimes," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 56-64.
    18. Gwo Dong Lin & Xiaoling Dou & Satoshi Kuriki, 2019. "The Bivariate Lack-of-Memory Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(2), pages 273-297, December.
    19. Li, Xiaohu & Fang, Rui, 2015. "Ordering properties of order statistics from random variables of Archimedean copulas with applications," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 304-320.
    20. Yinping You & Xiaohu Li & Narayanaswamy Balakrishnan, 2014. "On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(8), pages 1041-1056, November.
    21. Li, Chen & Li, Xiaohu, 2021. "On stochastic dependence in residual lifetime and inactivity time with some applications," Statistics & Probability Letters, Elsevier, vol. 177(C).

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