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Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

Part of: Applications Markov processes Parametric inference

Published online by Cambridge University Press:  01 July 2016

N. Lalam  and
C. Jacob
N. Lalam*
Affiliation:
Institut National de la Recherche Agronomique, Jouy-en-Josas
C. Jacob*
Affiliation:
Institut National de la Recherche Agronomique, Jouy-en-Josas
*
∗ Current address: Eurandom, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗ Postal address: INRA, Laboratoire de Biométrie, 78352 Jouy-en-Josas Cedex, France. Email address: [email protected]
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Abstract

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β < 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

Keywords

Size-dependent branchingconditional least squarespolymerase chain reaction

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62F12: Asymptotic properties of estimators
Secondary: 62P10: Applications to biology and medical sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 582 - 601
Copyright
Copyright © Applied Probability Trust 2004 

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