On inference from Markov chain macro-data using transforms

We consider data that are longitudinal, arising from n individuals over m time periods. Each individual moves according to the same homogeneous Markov chain, with s states. If the individual sample paths are observed, so that ‘micro-data’ are available, the transition probability matrix is estimated by maximum likelihood straightforwardly from the transition counts. If only the overall numbers in the various states at each time point are observed, we have ‘macro-data’, and the likelihood function is difficult to compute. In that case a variety of methods has been proposed in the literature. In this paper we propose methods based on generating functions and investigate their performance.