Exponential Superimposed Renewal Processes

Superimposed renewal processes arise when the events from multiple renewal processes and all labels, which allow us to identify where they have come from, are missing. In reliability engineering, individual renewal processes are stochastic failure time models for components, and the superimposed renewal process from them is the model for the system failure times. Labels are the identity information of individual failures, such as the serial number of the failed component, or the most recent failure time. When these labels are missing, it is challenging to conduct statistical inference on renewal processes because of the loss of information on identities. Recently a likelihood-based method is proposed under the assumption that all renewal processes under a superimposed process are homogeneous. This paper discusses the argument used in the method and tries to extend it for heterogeneous cases.