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stochastic orders; positive dependence orders; residual lifetimes; NBU; IFR; bivariate aging; survival copulas
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].
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