Abstract

Renewal processes are widely used to model regularly occurring events such as, e.g., lifetimes of technical components having to be replaced by identical units upon failure, waiting times between two successive incoming claims in an insurance company or consecutive production times in a plant. Suppose we inspect such a renewal process at a given time t, then there are several possibilities for what quantities may be observed. In the first case and in our example with technical components, one may have observed all lifetimes up to the last component failure before time t; another possibility is having observed the age of the component in use at time t additionally; lastly, only the number of failures up to time t may have been observed.
There are several different statistical aspects and procedures in the literature that have been applied to this context of renewal theory, some of which we will give a short introduction to. Also, we will briefly discuss respective results from the literature regarding specific phenomena in renewal theory such as the inspection paradox, the residual lifetime and age. Then, instead of a fixed time t, we will consider the renewal process in connection with a so-called random time of inspection which is assumed to be independent of the underlying renewal process. By this approach, we obtain some insights regarding statistical procedures for the number of renewals up to the random inspection time, utilize a connection with exponential families in a particular case and give several examples.