ESTIMATION THEORY FOR GROWTH AND IMMIGRATION RATES IN A MULTIPLICATIVE PROCESS

This paper deals with the simple Galton-Watson process with immigration, {X n } with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F’(l−) < 1), and that 0 < » ≡ B’(l−) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {X n } is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1 − m)−1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales ; and discusses relation of the above theory to that of a first order autoregressive process.

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