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

In many situations, the data given on a p-type Galton-Watson process Z_n eP N^p will consist of the total generation sizes |Z_n| only. In that case, the maximum likelihood estimator ρ_ML of the growth rate ρ is not observable, and the asymptotic properties of the most obvious estimators of ρ based on the |Z_n|, 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 ρ_ML. 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 ρ_ML and based on the |Z_n| alone is exhibited for the case p= 2, ρ₁²≥ρ.