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.