Further Monotonicity Properties for Specialized Renewal Processes

Define $Z(t)$ to be the forward recurrence time at t for a renewal process with interarrival time distribution, F, which is assumed to be IMRL (increasing mean residual life). It is shown that Eφ(Z(t)) is increasing in t ≥ 0 for all increasing convex φ. An example demonstrates that $Z(t)$ is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for F DFR (decreasing failure rate).