Let $𝐗=(X,Y)$ be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let ${𝐗}_t=\left[(X-t,Y-t)\right|X>t,Y>t]$ denotes the corresponding pair of residual lifetimes after time $t$, with $t\ge 0$. This note deals with stochastic comparisons between $𝐗$ 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...

We present optimal upper bounds for expectations of order statistics from i.i.d. samples with a common distribution function belonging to the restricted family of probability measures that either precede or follow a given one in the star ordering. The bounds for families with monotone failure density and rate on the average are specified. The results are obtained by projecting functions onto convex cones of Hilbert spaces.