Borovitskaja A. Asymptotic properties of intensity function estimators of Poisson processes and fields

The thesis is devoted to construction and investigation of asymptotic behaviour of maximum likelihood estimates of the intensity of Poisson fields. The nonparametric estimate of the intensity function of a nonhomogeneous Poisson field is constructed. The consistency of the estimate is studied and the rate of mean-square convergence of the estimate is investigated. Conditions for the asymptotic normality of smooth functionals of the estimate are obtained. For the maximum likelihood estimate of the compensator of a Poisson field, the bound for the rate of convergence in the uniform norm is given. Properties of maximum likelihood estimates of the intensity function of a nonhomogeneous marked Poisson field are investigated. The consistency of the nonparametric estimate and the rate of mean-square convergence of the estimate are studied.The consistency and asymptotic normality of the parametric estimate are investigated.