Estimation of the parameters of superpositions of Poisson point processes. (English. Russian original) Zbl 0748.62047

Molchanov, I. S.

Theory Probab. Math. Stat. 43, 123-130 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 111-117 (1990).

Summary: Estimators are obtained for the intensity of a stationary Poisson process \(\prod_ \lambda\) and for the set \(\tilde M=\{a_ i-a_ j: 1\leq i,\;j\leq m\}\) from observations of the superposition of the Poisson processes \(\prod_ \lambda+a_ i\), \(i=1,2,\dots,m\). Here \(\{a_ 1,\dots,a_ m\}\) is a nonrandom point set in \(R^ d\). It is proved that these estimators are strictly consistent, in the Hausdorff metric.

MSC:

62M09	Non-Markovian processes: estimation
62F12	Asymptotic properties of parametric estimators
62M05	Markov processes: estimation; hidden Markov models
60D05	Geometric probability and stochastic geometry
60G10	Stationary stochastic processes

Keywords:

strict consistency; superpositions of Poisson processes; Boolean models; intensity; stationary Poisson process; Hausdorff metric