This is part II in a series on residual life, hazard rates, and long-tailed distributions. If you haven’t read part I yet, read that first! The previous post in this series highlighted that one must be careful in connecting “heavy-tailed” with the concepts of “increasing mean residual life” and “decreasing hazard rate.”

In particular, there are many examples of light-tailed distributions that are IMRL and DHR. However, if we think again about the informal examples that we discussed in the previous post, it becomes clear that IMRL and DHR are too “precise” to capture the phenomena that we were describing. For example, if we return to the case of waiting for a response to an email, it is not that we expect our remaining waiting time to be monotonically increasing as we wait. If fact, we are very likely to get a response quickly, so the expected waiting time should drop initially (and the hazard rate should increase initially). It is only after we have waited a “long” time already, in this case a few days, that we expect to see a dramatic increase in our residual life. Further, in the extreme, if we have not received a response in a month, we can reasonably expect that we may never receive a response, and so the mean residual life is, in some sense, growing unboundedly, or equivalently, the hazard rate is decreasing to zero. The example of waiting for a subway train highlights the same issues. Initially, we expect that the mean residual life should decrease, because if the train is on schedule, things are very predictable. However, once we have waited a long time beyond when the train was supposed to arrive, it likely means something went wrong, and could mean the train has had some sort of mechanical problem and will never arrive.