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On The Role of a Certain Eigenvalue in Estimating the Growth Rate of a Branching Process

Summary 
In many situations, the data given on a p-type Galton-Watson process Zn eP Np will consist of the total generation sizes |Zn| only. In that case, the maximum likelihood estimator ρML of the growth rate ρ is not observable, and the asymptotic properties of the most obvious estimators of ρ based on the |Zn|, as studied by Asmussen & Keiding (1978), show a crucial dependence on |ρ1|,ρ1 being a certain other eigenvalue of the offspring mean matrix. In fact, if |ρ1|2≤ρ, then the speed of convergence compares badly with ρML. In the present note, it is pointed out that recent results of Heyde (1981) on so-called Fibonacci branching processes provide further examples of this phenomenon, and an estimator with the same speed of convergence as ρML and based on the |Zn| alone is exhibited for the case p= 2, ρ12≥ρ.