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

Consider two random vectors X 1 and X 2 whose distributions are defined according to the multivariate frailty approach, and let X k , t = [ X k − t | X k > t ] , k = 1 , 2 , be the corresponding vectors of residual lifetimes at t = ( t 1 , … , t n ) , t i ∈ 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 X 1 , t ≤ st X 2 , t , where t is an arbitrary vector in R n . Sufficient conditions for the stochastic comparisons X i , t ≤ st X i , t + v , where t is as above and v is a vector with non-negative components, are presented too.