On stochastic comparisons of excess times

A stationary renewal process based on iid random variables $X_i$ is observed at a given time. The excess time, that is, the residual time until the next renewal event, is of course smaller than the total current $X$ which consists of the residual time plus the current age. Nevertheless in certain types of data the distribution of the excess times is stochastically larger than that of $X_i$'s. We find necessary and sufficient conditions that explain this phenomenon, and related results on stochastic orderings arising from observations on renewal processes.