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On multiple change-point estimation for Poisson process. (English) Zbl 1390.62165

Summary: This work is devoted to the problem of change-point parameter estimation in the case of the presence of multiple changes in the intensity function of the Poisson process. It is supposed that the observations are independent inhomogeneous Poisson processes with the same intensity function and this intensity function has two jumps separated by a known quantity. The asymptotic behavior of the maximum-likelihood and Bayesian estimators are described. It is shown that these estimators are consistent, have different limit distributions, the moments converge and that the Bayesian estimators are asymptotically efficient. The numerical simulations illustrate the obtained results.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F15 Bayesian inference
62F12 Asymptotic properties of parametric estimators

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References:

[1] Akman, V. E., and A. E. Raftery. 1986. Asymptotic inference for a change-point Poisson process. Annals of Statistics 14 (4):1583-90. · Zbl 0648.62093
[2] Basseville, M., and I. V. Nikiforov. 1993. Detection of Abrupt changes: theory and application.Englewood Cliffs, N.J.: Prentice-Hall. · Zbl 1407.62012
[3] Chernov, H., and H. Rubin. 1956. The estimation of the location of a discontinuity in density. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 1:19-38.
[4] Deshayes, J.1983. Ruptures de modèles en statistique. Thèse d’État, Université Paris-Sud.
[5] Gal’tchouk, L. I., and B. L. Rozovskii. 1971. The disorder problem for a Poisson process. Theory of Probability and its Applications 16:712-6.
[6] Galun, S. A., and A. P. Trifonov. 1982. Detection and estimation of the time when the Poisson flow intensity changes. Automation and Remove Control 43 (6):782-90. · Zbl 0506.93056
[7] Gyarmati-Szaba, J., L. V. Bogachev, and H. Chena. 2011. Modelling threshold exceedances of air pollution concentrations via non-homogeneous Poisson process with multiple change-points. Atmospheric Environment 45 (31):5493-503.
[8] Gagliardi, R. M., and S. Karp. 1995. Optical communications. 2nd ed. New York: Wiley.
[9] Ibragimov, I. A., and R. Z. Khasminskii. 1972. Asymptotic behavior of statistical estimates for samples with a discontinuous density. Mathematics in USSR-Sbornik 16 (4):573-606.
[10] Ibragimov, I. A., and R. Z. Khasminskii. 1975. Parameter estimation for a discontinuous signal in white Gaussian noise. Problems of Information Transmission 11:203-12.
[11] Ibragimov, I. A., and R. Z. Khasminskii. 1981. Statistical estimation. Asymptotic theory.New York: Springer.
[12] Kutoyants, Yu. A.1984. Parameter estimation for stochastic processesHeldermann-Verlag: Berlin.
[13] Kutoyants, Yu. A.1998. Statistical inference for spatial Poisson processesN.Y.: Springer. · Zbl 0904.62108
[14] Pflug, G. C.1983. The limiting log-likelihood process for discontinuous density families. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 64:15-35. · Zbl 0525.62034
[15] Poor, V., and O. Hadjiliadis2009. Quickest detectionCambridge: Cambridge University Press.
[16] Rubin, H.1961. The estimation of the discontinuities in multivariate densities, and related problems in stochastic process. Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability 1:563-74.
[17] Shiryaev, A. N.1963. On optimum methods in quickest detection problems. Theory of Probability and its Applications 8:22-46. · Zbl 0213.43804
[18] Shiryaev, A. N.2008. Optimal stopping rules.Berlin: Springer. · Zbl 1138.60008
[19] Trifonov, A. P., and T. M. Ovchinnikova. 1989. Reception of an optical signal with unknown delay. Radioellectronics and Communications Systems 32 (8):24-28.
[20] Van Trees, H. L., K. L. Bell, and Z. Tian. 2013. Detection, estimation and modulation theory. Part I, N.Y.: John Wiley & Sons. · Zbl 1266.94001
[21] West, W. R., and T. R., Ogden1997. Continuous-time estimation of a change-point in a Poisson process. Journal of Statistical Computation and Simulation 56 (4):293-302. · Zbl 0872.62082
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