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On The Role of a Certain Eigenvalue in Estimating the Growth Rate of a Branching Process
Author(s) -
Asmussen Søren
Publication year - 1982
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1982.tb00819.x
Subject(s) - mathematics , estimator , fibonacci number , eigenvalues and eigenvectors , branching process , branching (polymer chemistry) , rate of convergence , observable , combinatorics , statistics , physics , computer science , chemistry , quantum mechanics , computer network , channel (broadcasting) , organic chemistry
Summary 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 |ρ 1 |,ρ 1 being a certain other eigenvalue of the offspring mean matrix. In fact, if |ρ 1 | 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, ρ 1 2 ≥ρ.