Predicting Extinction or Explosion in a Galton-Watson Branching Process with Power Series O↵spring Distribution⇤

Extinction is certain in a Galton-Watson (GW) branching process if the o↵spring mean µ < 1, whereas explosion is possible but not certain if µ> 1. Discriminating between these two possibilities is a well-studied hypothesis-testing problem. However, deciding whether extinction or explosion will occur for the current realization of the process is a prediction problem. This can be formulated as a di↵erent testing problem by considering the conditional distributions of the process given extinction and explosion respectively. For power series o↵spring distributions, fixed-sample and sequential parametric tests are presented for the prediction problem and illustrated with data on the spread of epidemics and the populations of endangered species. ⇤Key words and phrases: Galton-Watson branching process; extinction; explosion; sub-critical; supercritical; stochastic ordering; prediction; hypothesis testing; least favorable distribution; sequential probability ratio test; epidemic; endangered species