Negative aging and stochastic comparisons of residual lifetimes in multivariate frailty models

Consider two random vectors X1X1 and X2X2 whose distributions are defined according to the multivariate frailty approach, and let Xk,t=[Xk−t|Xk>t]Xk,t=[Xk−t|Xk>t], k=1,2k=1,2, be the corresponding vectors of residual lifetimes at t=(t1,…,tn),ti∈R,i=1,…,n. Conditions for multivariate stochastic comparisons of random vectors described by the frailty approach have been recently presented in Misra et al. (2009). Here we prosecute their study, providing sufficient conditions for the stochastic comparison X1,t≤stX2,tX1,t≤stX2,t, where tt is an arbitrary vector in RnRn. Sufficient conditions for the stochastic comparisons Xi,t≤stXi,t+vXi,t≤stXi,t+v, where tt is as above and vv is a vector with non-negative components, are presented too.