Ordering properties of the TTT-plot of lifetimes with Schur joint densities

In this note we study the law of scaled empirical TTT-plots of exchangeable lifetimes. By means of very simple arguments, we prove a monotonicity property in the case of absolutely continuous distributions with Schur-concave (or Schur-convex) densities. A stochastic comparison between the laws of TTT-plots for i.i.d. exponential variables and for Schur-concave (or Schur-convex) densities follows as a direct consequence. This can be seen as a natural analogue of a well-known fact valid for the case of i.i.d. lifetimes with monotone failure rates. It is remarkable that a stochastic comparison is even obtained in the likelihood ratio ordering sense.