Statistical inference in branching processes

This paper surveys the theoretical literature on statistical inference for branching processes, almost exclusively from the last five years. The emphasis is on estimation of the offspring mean (time-discrete case) and the Malthusian growth parameter (time-continuous case). Essentially two different repetitive structures are important: increasing number of ancestors, leading to standard i.i.d. theory, and increasing number of generations with which we shall be concerned. Very little thought has been given to small-sample theory, including suitable conditionality arguments for defining reasonable reference sampling distributions. For the large sample theory for supercritical branching processes in discrete time it seems useful to view a series of generations as successive sections (of random length) of i.i.d. replications of a random variable with the offspring distribution. The continuous-time analogue is the random time transformation to a Poisson process (which has i.i.d. increments) via the inverse of the natural increasing process corresponding to the split time process, this being a trivial submartingale. The latter approach has been used by Aalen for statistical analysis of more general point processes. The use of the natural increasing process as a measure of efficiency has been studied for both discrete and continuous time by a series of Australian authors, who have also provided suggestions as to definition of efficiency concerning tests of hypotheses on the offspring mean. Essentially, these concepts reduce consideration to the underlying structure of i.i.d. birth events, but the general formulation suggests wider applicability within the theory of statistical inference for stochastic processes. The large sample results for the estimators state strong consistency on the set of non-extinction and asymptotic normality, using the above mentioned natural increasing process (total exposure time) as random normalising factor. A unifying tool is martingale central limit theory. Other areas are the subcritical process with immigration, for which some time-series analogue estimators have been investigated by martingale methods. Very recently, also multitype processes have been studied. For this case as well as for Bellman-Harris processes in continuous time, the rate of convergence of the estimator towards the parameter turns out to be critically dependent on the rate of convergence to the stable type (or age) distribution.