Comptes Rendus
Statistics/Probability Theory
Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 663-666.

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m1 with a rate of convergence of order Nnα as the population size Nn grows to ∞ and the variance may change at the rate Nnβ, where α>0 and 1β<1. The offspring mean depends on an unknown parameter θ0 that we estimate on the non-extinction set by using the conditional least squares method. We prove the strong consistency of the estimator of θ0 as n under some general conditions on the asymptotic behavior of the process. We also give its asymptotic distribution for a certain class of size-dependent branching processes.

On considère un processus de branchement taille-dépendant unitype {Nn}n supercritique ou presque-critique tel que sa descendance moyenne converge vers une limite m1 à une vitesse de l'ordre de Nnα lorsque l'effectif de la population Nn tend vers l'infini et tel que sa variance évolue à la vitesse Nnβα>0 et 1β<1. La descendance moyenne dépend d'un paramètre inconnu θ0 que l'on estime sur l'ensemble de non-extinction du processus à l'aide de la méthode des moindres carrés conditionnels. On démontre la consistance forte de l'estimateur de θ0 quand n sous des hypothèses générales concernant le comportement asymptotique du processus. On donne aussi sa distribution asymptotique pour une certaine classe de processus taille-dépendants.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.001
Nadia Lalam 1; Christine Jacob 2

1 EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2 Unité de biométrie, INRA, 78352 Jouy-en-Josas cedex, France
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Nadia Lalam; Christine Jacob. Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 663-666. doi : 10.1016/j.crma.2004.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.001/

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[3] G. Kersting Some properties of stochastic difference equations (P. Tautu, ed.), Stochastic Modelling in Biology, World Scientific, Singapore, 1990, pp. 328-339

[4] F.C. Klebaner On population-size dependent branching processes, Adv. Appl. Probab., Volume 16 (1984), pp. 30-55

[5] T.L. Lai Asymptotic properties of nonlinear least squares estimates in stochastic regression models, Ann. Statist., Volume 22 (1994), pp. 1917-1930

[6] N. Lalam, C. Jacob, Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process, Technical report, 2003

[7] I. Rahimov Random Sums and Branching Stochastic Processes, Lecture Notes in Statist., Springer-Verlag, New York, 1995

[8] K. Skouras Strong consistency in nonlinear stochastic regression models, Ann. Statist., Volume 28 (2000), pp. 871-879

[9] C.F. Wu Asymptotic theory of nonlinear least squares estimation, Ann. Statist., Volume 9 (1981), pp. 501-513

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