Estimation theory for growth and immigration rates in a multiplicative process

Abstract

This paper deals with the simple Galton-Watson process with immigration, {Xn } 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'(1–) < 1), and that 0 < λ ≡ B'(1–) < ∞, 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 {Xn } 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.