Predicting extinction or explosion in a Galton–Watson branching process

Based on observations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1,\dots ,X_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo stretchy="false">…</mo> <mo>,</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> of successive generations of a discrete-parameter Galton–Watson branching process, one wishes to predict whether extinction or explosion will ultimately occur. This problem can be formulated as a simple hypothesis-testing problem to which the Neyman–Pearson Lemma is directly applicable if the extinction probability is known or estimable. If it is not, valid (but conservative) tests still can be obtained. Copyright Springer Science+Business Media Dordrecht 2013