Estimating the distribution and thinning parameters of a homogeneous multimode Poisson process

In this paper we propose estimators of the distribution of events of different kinds in a multimode Poisson process. We give the explicit solution for the maximum likelihood estimator, and derive its strong consistency and asymptotic normality. We also provide an order restricted estimator and derive its consistency and asymptotic distribution. We discuss the application of the estimator to the detection of neutrons in a novel detector being developed at the European Spallation Source in Lund, Sweden. The inference problem gives rise to Sylvester-Ramanujan system of equations.