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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
Summary:
Let $\mbox{\boldmath$X$} = (X,Y)$ be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let $\mbox{\boldmath$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 $\mbox{\boldmath$X$}$ and $\mbox{\boldmath$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].
References:
[1] Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York 1966 MR 0208210
[2] Avérous J., Dortet-Bernadet M. B.: Dependence for Archimedean copulas and aging properties of their generating functions. Sankhya 66 (2004), 607–620 MR 2205812
[3] Barlow R. E., Proschan F.: Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD 1981 Zbl 0379.62080
[4] Bassan B., Spizzichino E.: Bivariate survival models with Clayton aging functions. Insurance: Mathematics and Economics 37 (2005), 6–12 DOI 10.1016/j.insmatheco.2004.12.003 | MR 2156592 | Zbl 1075.62091
[5] 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
[6] Charpentier A.: Tail distribution and dependence measure. In: Proc. 34th ASTIN Conference 2003
[7] Charpentier A.: Dependence Structures and Limiting Results, with Applications in Finance and Insurance. Ph.D. Thesis, Katholieke Universiteit of Leuven 2006
[8] Clayton D. G.: A model for association in bivariate life tables and its applications in epidemiological studies of familiar tendency in chronic disease incidence. Biometrika 65 (1978), 141–151 DOI 10.1093/biomet/65.1.141 | MR 0501698
[9] Cook R. D., Johnson M. E.: A family of distributions for modelling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. Ser. B 43 (1981), 210–218 MR 0626768 | Zbl 0471.62046
[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
[11] Genest C., MacKay R. J.: Copules archimédiennes et familles de lois bidimensionelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145–159 DOI 10.2307/3314660 | MR 0849869
[12] Ghurye S. G., Marshall A. W.: Shock processes with aftereffects and multivariate lack of memory. J. Appl. Probab. 21 (1984), 786–801 DOI 10.2307/3213696 | MR 0766816 | Zbl 0553.60078
[13] Juri A., Wüthrich M. V.: Copula convergence theorems for tail events. Insurance: Mathematics and Economics 30 (2002), 405–420 DOI 10.1016/S0167-6687(02)00121-X | MR 1921115 | Zbl 1039.62043
[14] Marshall A. W., Shaked M.: A class of multivariate new better than used distributions. Ann. Probab. 10 (1982), 259–264 DOI 10.1214/aop/1176993931 | MR 0637394 | Zbl 0481.62077
[15] Nelsen R. B.: An Introduction to Copulas. Springer, New York 1999 MR 1653203 | Zbl 1152.62030
[16] Oakes D.: On the preservation of copula structure under truncation. Canad. J. Statist. 33 (2005), 465–468 DOI 10.1002/cjs.5540330310 | MR 2193986 | Zbl 1101.62040
[17] Pellerey F., Shaked M.: Characterizations of the IFR and DFR aging notions by means of the dispersive order. Statist. Probab. Lett. 33 (1997), 389–393 DOI 10.1016/S0167-7152(96)00152-6 | MR 1458009 | Zbl 0903.60081
[18] Shaked M., Shanthikumar J. G.: Stochastic Orders. Springer, New York 2007 MR 2265633 | Zbl 0883.60016
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