Geodesic
On the estimation of the intensity density function of Poisson random field outside of the observation region
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 97-109 
I. A. Ibragimov. On the estimation of the intensity density function of Poisson random field outside of the observation region. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 97-109. http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a6/
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     author = {I. A. Ibragimov},
     title = {On the estimation of the intensity density function of {Poisson} random field outside of the observation region},
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     year = {2014},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a6/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A Poisson random field with the intensity density function $\frac{\lambda(x)}\varepsilon$ is observed in a bounded region $G\subseteq\mathbb R^d$. It is supposed that the unknown function $\lambda$ belongs to a known class of entire functions. The parameter $\varepsilon$ is supposed to be known. The problem is to estimate the value $\lambda(x)$ at the points $x\notin G$. We consider an asymptotic setup of the problem when $\varepsilon\to0$.

[1] S. N. Bernshtein, Ekstremalnye svoistva polinomov, ONTI, M.-L., 1937

[2] I. A. Ibragimov, “Ob ekstrapolyatsii tselykh funktsii nablyudaemykh v gaussovskom belom shume”, Ukrainskii matem. zhurn., 21:3 (2000), 58–69

[3] I. A. Ibragimov, “Ob otsenivanii analiticheskoi plotnosti raspredeleniya po tsenzurirovannoi vyborke”, Zap. nauchn. semin. POMI, 311, 2004, 147–160 | MR | Zbl

[4] I. A. Ibragimov, “Ob otsenke mnogomernoi analiticheskoi plotnosti raspredeleniya po tsenzurirovannoi vyborke”, Teoriya veroyatn. i ee primen., 51:1 (2006), 95–108 | DOI | MR | Zbl

[5] I. A. Ibragimov, “O nizhnikh granitsakh tochnosti neparametricheskogo otsenivaniya”, Zap. nauchn. semin. POMI, 396, 2011, 102–110 | MR

[6] I. A. Ibragimov, R. Z. Khasminskii, Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979 | MR

[7] I. A. Ibragimov, R. Z. Khasminskii, “Ob otsenke plotnosti raspredeleniya, prinadlezhaschei odnomu klassu tselykh funktsii”, Teoriya veroyatn. i ee primen., 27:3 (1982), 514–524 | MR | Zbl

[8] Dzh. Kingman, Puassonovskie protsessy, MTsNMO, M., 2007

[9] Yu. A. Kutoyants, Statistical Inference for Spatial Poisson Processes, Lecture Notes Statist., 134, Springer, 1998 | DOI | MR | Zbl

[10] G. Sege, Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[11] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[12] R. Hasminskii, I. Ibragimov, “On density estimation in the view of Kolmogorov's ideas in approximation theory”, Ann. Statist., 18:3 (1990), 999–1010 | DOI | MR | Zbl

[13] N. N. Chentsov, “Otsenka neizvestnoi plotnosti raspredeleniya po nablyudeniyam”, Dokl. AN SSSR, 147:1 (1962), 45–48 | MR | Zbl