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Inference for immigration-death processes with single and paired immigrants

The problem of parameter estimation for immigration-death models when only death times are observed is considered. For the simplest immigration-death model an analytic solution for the likelihood exists and this is used to investigate sampling properties of profile likelihoods. The analysis is extended to more complex models (allowing immigration of pairs) that are of interest in quantum optics. Parameter posterior densities, previously investigated by the authors using Markov chain methods, are investigated and the potentially misleading conclusions produced by Markov chain samplers for this application are highlighted.