On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas

Franco Pellerey

Kybernetika (2008)

  • Volume: 44, Issue: 6, page 795-806
  • ISSN: 0023-5954

Abstract

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Let 𝐗 = ( X , Y ) be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let 𝐗 t = [ ( X - t , Y - t ) | X > t , Y > t ] denotes the corresponding pair of residual lifetimes after time t , with t ≥ 0 . This note deals with stochastic comparisons between 𝐗 and 𝐗 t : we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate and bivariate aging. This work is mainly based on, and related to, recent papers by Bassan and Spizzichino ([4] and [5]), Averous and Dortet-Bernadet [2], Charpentier ([6] and [7]) and Oakes [16].

How to cite

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Pellerey, Franco. "On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas." Kybernetika 44.6 (2008): 795-806. <http://eudml.org/doc/33965>.

@article{Pellerey2008,
abstract = {Let $\mbox\{$X$\} = (X,Y)$ be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let $\mbox\{$X$\}_t= [(X-t,Y-t) \vert X>t, Y>t]$ denotes the corresponding pair of residual lifetimes after time $t$, with $t \ge 0$. This note deals with stochastic comparisons between $\mbox\{$X$\}$ and $\mbox\{$X$\}_t$: we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate and bivariate aging. This work is mainly based on, and related to, recent papers by Bassan and Spizzichino ([4] and [5]), Averous and Dortet-Bernadet [2], Charpentier ([6] and [7]) and Oakes [16].},
author = {Pellerey, Franco},
journal = {Kybernetika},
keywords = {stochastic orders; positive dependence orders; residual lifetimes; NBU; IFR; bivariate aging; survival copulas; stochastic orders; positive dependence orders; residual lifetimes; NBU; IFR; bivariate aging; survival copulas},
language = {eng},
number = {6},
pages = {795-806},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas},
url = {http://eudml.org/doc/33965},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Pellerey, Franco
TI - On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas
JO - Kybernetika
PY - 2008
PB - Institute of Information Theory and Automation AS CR
VL - 44
IS - 6
SP - 795
EP - 806
AB - Let $\mbox{$X$} = (X,Y)$ be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let $\mbox{$X$}_t= [(X-t,Y-t) \vert X>t, Y>t]$ denotes the corresponding pair of residual lifetimes after time $t$, with $t \ge 0$. This note deals with stochastic comparisons between $\mbox{$X$}$ and $\mbox{$X$}_t$: we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate and bivariate aging. This work is mainly based on, and related to, recent papers by Bassan and Spizzichino ([4] and [5]), Averous and Dortet-Bernadet [2], Charpentier ([6] and [7]) and Oakes [16].
LA - eng
KW - stochastic orders; positive dependence orders; residual lifetimes; NBU; IFR; bivariate aging; survival copulas; stochastic orders; positive dependence orders; residual lifetimes; NBU; IFR; bivariate aging; survival copulas
UR - http://eudml.org/doc/33965
ER -

References

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  3. Barlow R. E., Proschan F., Statistical Theory of Reliability and Life Testing, To Begin With, Silver Spring, MD 1981 Zbl0379.62080
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  6. Charpentier A., Tail distribution and dependence measure, In: Proc. 34th ASTIN Conference 2003 
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  10. Foschi R., Spizzichino F., Semigroups of Semicopulas and Evolution of Dependence at Increase of Age, Technical Report, Department of Mathematics, University “La Sapienza”, Rome 2007 
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  13. Juri A., Wüthrich M. V., 10.1016/S0167-6687(02)00121-X, Insurance: Mathematics and Economics 30 (2002), 405–420 Zbl1039.62043MR1921115DOI10.1016/S0167-6687(02)00121-X
  14. Marshall A. W., Shaked M., 10.1214/aop/1176993931, Ann. Probab. 10 (1982), 259–264 (1982) Zbl0481.62077MR0637394DOI10.1214/aop/1176993931
  15. Nelsen R. B., An Introduction to Copulas, Springer, New York 1999 Zbl1152.62030MR1653203
  16. Oakes D., 10.1002/cjs.5540330310, Canad. J. Statist. 33 (2005), 465–468 Zbl1101.62040MR2193986DOI10.1002/cjs.5540330310
  17. Pellerey F., Shaked M., 10.1016/S0167-7152(96)00152-6, Statist. Probab. Lett. 33 (1997), 389–393 (1997) Zbl0903.60081MR1458009DOI10.1016/S0167-7152(96)00152-6
  18. Shaked M., Shanthikumar J. G., Stochastic Orders, Springer, New York 2007 Zbl0883.60016MR2265633

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