Recursive estimation of parameters in Markov-modulated Poisson processes

A hidden Markov regime is a Markov process that governs the time or space dependent distributions of an observed stochastic process. Recursive algorithms can be used to estimate parameters in mixed distributions governed by a Markov regime. The authors derive a recursive algorithm for estimation of parameters in a Markov-modulated Poisson process also called a Cox point process. By this the authors mean a doubly stochastic Poisson process with a time dependent intensity that can take on a finite number of different values. The intensity switches randomly between the possible values according to a Markov process. The authors consider two different ways to observe the Markov-modulated Poisson process: in the first model the observations consist of the observed time intervals between events, and in the second model they use the total number of events in successive intervals of fixed length. They derive an algorithm for recursive estimation of the Poisson intensities and the switch intensities between the two states and illustrate the algorithm in a simulation study. The estimates of the switch intensities are based on the observed conditional switch probabilities.