Pseudo-likelihood estimation for discretely observed multitype Bellman–Harris branching processes

This paper presents a method for parametric estimation in the class of multitype Bellman–Harris branching processes when the data consist of cell counts collected on several colonies observed at discrete time points. This sampling scheme arises frequently in biology to analyze cell proliferation in tissue culture. We investigate the use of a pseudo-likelihood approach in this context. It is defined from the mean vectors and variance–covariance matrices of the numbers of cells. The proposed estimator is strongly consistent and asymptotically normal as the number of observed colonies goes to infinity. In situations where these two moments have no closed-form expressions their values can be replaced by simulation-based approximations. The resulting simulated pseudo-maximum likelihood estimator is asymptotically equivalent to the pseudo-maximum likelihood estimator as the number of simulation increases. We illustrate the proposed method via two examples, and evaluate its finite sample properties through simulations.