Estimation for branching processes with varying and random environments

Abstract

We consider statistical estimation problems for discrete and continuous time Markov branching processes with both varying and random environments. Estimators are proposed for parametric functions of parameters by using martingale transforms in discrete time and stochastic integrals with respect to martingales in continuous time. In discrete time, we study estimators for time averages of reproduction probabilities and reproduction means; time averages of instantaneous birth rates are examined in continuous time. Asymptotic properties are studied for all the estimators proposed. For deterministically varying environments, asymptotic unbiasedness and consistency are proven, and asymptotic distributions are computed for increasing initial population sizes X 0 . When the environments are random, the estimators are shown to converge almost surelyto the true means of the environmental parameters as the time interval of observation becomes large and X 0 remains fixed.