Predicting extinction or explosion in a Galton–Watson branching process with power series offspring distribution

•Predicting extinction or explosion of one realization of a branching process.•This differs from the usual problem of testing the criticality parameter.•Both fixed sample size and sequential sampling methods are presented.•The sequential probability ratio test is simple and efficient for prediction.

Extinction is certain in a Galton–Watson (GW) branching process if the offspring mean μ≤1μ≤1, whereas explosion is possible but not certain if μ>1μ>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 different testing problem by considering the conditional distributions of the process given extinction and explosion respectively. For power series offspring 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.