Geodesic
On properties of estimators in nonregular situations for Poisson processes
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–1, Tome 363 (2009), pp. 26-47 
Y. A. Kutoyants. On properties of estimators in nonregular situations for Poisson processes. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–1, Tome 363 (2009), pp. 26-47. http://geodesic.mathdoc.fr/item/ZNSL_2009_363_a2/
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     title = {On properties of estimators in nonregular situations for {Poisson} processes},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_363_a2/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We consider the problem of parameter estimation by observations of inhomogeneous Poisson process. It is well-known that if the regularity conditions are fulfilled then the maximum likelihood and bayesian estimators are consistent, asymptotically normal, and asymptotically efficient. These regularity conditions can be roughly presented as follows: a) the intensity function of observed process belongs to known parametric family of functions, b) the model is identifiable, c) the Fisher information is positive continuous function, d) the intensity function is sufficiently smooth with respect to the unknown parameter, e) this parameter is an interior point of the interval. We are interested in the properties of estimators when these regularity conditions are not fulfilled. More precisely, we preset a review of the results which correspond to the rejection of these conditions one by one and we show how the properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii. Bibl. – 9 titles.

[1] M. V. Burnashev, Yu. A. Kutoyants, “On minimal $\alpha$-mean error parameter transmission over Poisson channel”, IEEE Transactions on Information Theory, 47:6 (2001), 2505–2515 | DOI | MR | Zbl

[2] A. Dabye, C. Farinetto, Yu. A. Kutoyants, “On Bayesian estimators in misspecified change-point problem for Poisson processes”, Stat. Prob. Lett., 61:1 (2003), 17–30 | DOI | MR | Zbl

[3] A. Dabye, Yu. A. Kutoyants, “On misspecified change-point problem for Poisson processes”, J. Appl. Prob., 38A (2001), 705–709 | DOI | MR

[4] S. Dachian, “Estimation of cusp location by Poisson observations”, SISP, 6:1 (2003), 1–14 | MR | Zbl

[5] I. A. Ibragimov, R. Z. Khasminskii, Statistical Estimation, Springer, New York, 1981 | MR | Zbl

[6] Yu. A. Kutoyants, Statistical Inference for Spatial Poisson Processes, Springer, New York, 1998 | MR | Zbl

[7] Yu. A. Kutoyants, “On regular and singular estimation for ergodic diffusion”, J. Japan Statist. Soc., 38 , Celebration Volume for H. Akaike (2008), 51–63 | DOI | MR

[8] Yu. A. Kutoyants, V. Spokoiny, “Optimal choice of observation window for Poisson observations”, Statist. Probab. Lett., 44 (1999), 291–298 | DOI | MR | Zbl

[9] N. Yoshida, T. Hayashi, “On the robust estimation in Poisson processes with periodic intensities”, AISM, 42:3 (1990), 489–507 | DOI | MR | Zbl