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Multiplier estimation for the distribution function of particle life time in a Bellman-Harris process. (Russian) Zbl 0618.60081

A Bellmann-Harris branching process is defined by a distribution function F \((F(0)=0)\) describing the life time of the particles and by a discrete distribution \({\mathcal P}:=\{p_ n\); \(n\geq 0\}\) of the number of offsprings of each particle. The life time and the number of offsprings are supposed to be independent for different particles. The paper is aimed at giving a nonparametric estimate of F if independent observations of the process are given. The j th observation occurs up to time \(T_ j\); \(T_ j\leq T_ 0<\infty\); \(j\geq 1\). \({\mathcal P}\) is considered as a nuisance parameter.
Convergence in probability of the estimate to the distribution function is proved. An appropriately standardized transform of the estimate is shown to converge weakly to the Wiener process in the Skorokhod space D[0,v].
Reviewer: J.Tóth

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
62G05 Nonparametric estimation
62M09 Non-Markovian processes: estimation