Asymptotic distributions for weighted estimators of the offspring mean in a branching process

Autores: I. Rahimov
Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 18, Nº. 3, 2009, págs. 568-583
Idioma: inglés
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Resumen
- It is known that conditional least squares estimator (CLSE) of the offspring mean for the process with a stationary immigration is not asymptotically normal. In the paper, we demonstrate that for the process with non-stationary immigration it may have a normal limit distribution. Considering a discrete time branching process Z(n) with time-dependent immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively, we show that 1+2α is the threshold for asymptotic normality of the estimator. It will be proved that if β<1+2α, the estimator is asymptotically normal with two different normalizing factors, and if β>1+2α its limiting distribution is not normal, but can be expressed in terms of certain functionals of the time-changed Wiener process. When β=1+2α, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. We derive all possible limit distributions of the weighted CLSE based on observations {Z(r+1),Z(r+2),…,Z(n)} as n→∞ and r=[n ε], 0≤ε<1. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.