Estimation theory for growth and immigration rates in a multiplicative process

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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 ratesmandλon the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for bothmandλare obtained, together with associated central limit theorems in relation tomandμ≡λ(1–m)–1Following 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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