Distributions of Renewal Variables When Renewal Process Reaches a Special Event

ABSTRACT

Motivated by a problem in estimating the incidence of a disease with delayed onset of symptoms, the current, excess, and total life length distributions of a renewal process at a random event time point are considered. Distributions of these renewal variables at a random time point can be easily derived if we know their corresponding distributions at a fixed time point. Unfortunately, the distribution conditional on a fixed time point is usually not available since it requires solving a renewal equation and an explicit solution of the renewal equation rarely exists. In this article, we consider the situation where the time to a specific event is random and independent of the renewal process. We assume that the occurrence rate of the event is constant, in other words, the time to the event for each individual follows an exponential distribution. Under these assumptions, we derive the distributions for the renewal variables considered. We also derive the distribution of the total life when excess life is bounded by an independent random variable. Using results developed in this article, one can derive the distributions of the renewal variables considered from the distribution of the renewal process and vice versa. Our results show that when the incidence rate is small, the renewal distributions can be approximated by the corresponding asymptotic distributions under equilibrium.

Keywords:

• Equilibrium
• Gamma distribution
• Incidence
• Renewal process
• Window period