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

Nadia Lalam; Christine Jacob

doi:10.1016/j.crma.2004.09.001

Statistics/Probability Theory

Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process
[Estimation de la descendance moyenne d'un processus de branchement taille-dépendant supercritique ou presque-critique.]

Présenté par : Paul Deheuvels

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 663-666.

Résumé
Abstract

On considère un processus de branchement taille-dépendant unitype ${N_{n}}_{n}$ supercritique ou presque-critique tel que sa descendance moyenne converge vers une limite $m ⩾ 1$ à une vitesse de l'ordre de $N_{n}^{α}$ lorsque l'effectif de la population $N_{n}$ tend vers l'infini et tel que sa variance évolue à la vitesse $N_{n}^{β}$ où $α > 0$ et $- 1 ⩽ β < 1$ . La descendance moyenne dépend d'un paramètre inconnu $θ_{0}$ 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 $θ_{0}$ quand $n \to \infty$ sous des hypothèses générales concernant le comportement asymptotique du processus. On donne aussi sa distribution asymptotique pour une certaine classe de processus taille-dépendants.

We consider a single-type supercritical or near-critical size-dependent branching process ${N_{n}}_{n}$ such that the offspring mean converges to a limit $m ⩾ 1$ with a rate of convergence of order $N_{n}^{α}$ as the population size $N_{n}$ grows to ∞ and the variance may change at the rate $N_{n}^{β}$ , where $α > 0$ and $- 1 ⩽ β < 1$ . The offspring mean depends on an unknown parameter $θ_{0}$ that we estimate on the non-extinction set by using the conditional least squares method. We prove the strong consistency of the estimator of $θ_{0}$ as $n \to \infty$ 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.

Reçu le : 2004-01-20
Accepté le : 2004-09-01
Publié le : 2004-10-27

DOI : 10.1016/j.crma.2004.09.001

Affiliations des auteurs :

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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     author = {Nadia Lalam and Christine Jacob},
     title = {Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {663--666},
     publisher = {Elsevier},
     volume = {339},
     number = {9},
     year = {2004},
     doi = {10.1016/j.crma.2004.09.001},
     language = {en},
}

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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/

Bibliographie
Cité par

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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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