Identification of multitype branching processes
Abstract
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\'esout le probl\`eme de construction d'une r\'egion 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\'ee sur un th\'eor\`eme de limite centrale associ\'e \`a une loi forte quadratique v\'erifi\'ee 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\'es de la matrice des moyennes est aussi bri\`evement comment\'ee.
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