  This paper is concerned with combined inference for point processes on the
real line observed in a broken interval. For such processes, the classic
history-based approach cannot be used. Instead, we adapt tools from sequential
spatial point processes. For a range of models, the marginal and conditional
distributions are derived. We discuss likelihood based inference as well as
parameter estimation using the method of moments, conduct a simulation study
for the important special case of renewal processes and analyse a data set
collected by Diggle and Hawtin.
