Identification of multitype branching processes

Mathematics – Statistics Theory

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Published at http://dx.doi.org/10.1214/009053605000000561 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000561

We solve the problem of constructing an asymptotic global confidence region for the means and the covariance matrices of the reproduction distributions involved in a supercritical multitype branching process. Our approach is based on a central limit theorem associated with a quadratic law of large numbers performed by the maximum likelihood or the multidimensional Lotka--Nagaev estimator of the reproduction law means. The extension of this approach to the least squares estimator of the mean matrix is also briefly discussed. On r\'{e}sout le probl\`{e}me de construction d'une r\'{e}gion de confiance asymptotique et globale pour les moyennes et les matrices de covariance des lois de reproduction d'un processus de branchement multitype et supercritique. Notre approche est bas\'{e}e sur un th\'{e}or\`{e}me de limite centrale associ\'{e} \`{a} une loi forte quadratique v\'{e}rifi\'{e}e par l'estimateur du maximum de vraisemblance ou l'estimateur multidimensionnel de Lotka--Nagaev des moyennes des lois de reproduction. L'extension de cette approche \`{a} l'estimateur des moindres carr\'{e}s de la matrice des moyennes est aussi bri\`{e}vement comment\'{e}e.

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