Asymptotic properties of nonparametric estimates of intensity of nonhomogeneous Poisson fields. (Ukrainian, English) Zbl 1023.62052

Kukush, O. G.; Stenanyshcheva, A. O.

Teor. Jmovirn. Mat. Stat. 65, 91-103 (2001); translation in Theory Probab. Math. Stat. 65, 101-114 (2002).

Summary: The authors deal with the problem of nonparametric estimation of the unknown intensity function \(\lambda(t)\) of a nonhomogeneous Poisson field. Under the condition that this intensity belongs to a compact set of a Sobolev space a maximum likelihood estimate is constructed based on observations of the field on an increasing sequence of sets. Conditions under which the proposed estimate is consistent are presented. The asymptotic normality of some functionals is proved. The rate of convergence in the uniform metric is estimated. A similar problem was considered by R. Helmers and R. Zitikis [Ann. Inst. Stat. Math. 51, 265-280 (1999; Zbl 0946.62076)].

MSC:

62G20	Asymptotic properties of nonparametric inference
62M09	Non-Markovian processes: estimation
62G05	Nonparametric estimation
62G07	Density estimation

Keywords:

nonhomogeneous Poisson field; intensity; asymptotic normality; rate of convergence; uniform metric; consistency

Citations:

Zbl 0946.62076