What can or can't be estimated in branching and related processes? (Q1263206)

This note compares the incompletely observed random walk and Galton- Watson branching tree. For the random walk, parameters not completely determined by the first few moments of the underlying distribution cannot be consistently estimated. A similar result holds for the branching tree with the observations now being sums of family sizes. In the case where the offspring distribution belongs to a power series family, it is possible to obtain the maximum likelihood estimators from an approximate likelihood. The estimates are the same as when individual family sizes are observed.