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homogeneous and non-homogeneous Poisson process; counting process; change point detection
The paper concentrates on modeling the data that can be described by a homogeneous or non-homogeneous Poisson process. The goal is to decide whether the intensity of the process is constant or not. In technical practice, e.g., it means to decide whether the reliability of the system remains the same or if it is improving or deteriorating. We assume two situations. First, when only the counts of events are known and, second, when the times between the events are available. Several statistical tests for a detection of a change in an intensity of the Poisson process are described and illustrated by an example. We cover both the case when the time of the change is assumed to be known or unknown.
[1] Antoch J., Hušková M.: Estimators of changes. In: Nonparametrics, Asymptotics an Time Series (S. Ghosh, ed.), M. Dekker, New York 1998, pp. 533–578 MR 1724708
[2] Antoch J., Hušková, M., Jarušková D.: Off-line quality control. In: Multivariate Total Quality Control: Foundations and Recent Advances (N. C. Lauro et al. eds.), Springer–Verlag, Heidelberg 2002, pp. 1–86 MR 1886415
[3] Barlow R. E., Proschan F.: Mathematical Theory of Reliability. Wiley, New York 1964 MR 0195566 | Zbl 0874.62111
[4] Chernoff H., Zacks S.: Estimating the current mean of normal distribution which is subjected to changes in time. Ann. Math. Statist. 35 (1964), 999–1018 MR 0179874
[5] Cox D. R., Lewis P. A. W.: The Statistical Analysis of Series of Events. Wiley, New York 1966 MR 0199942 | Zbl 0195.19602
[6] Csörgő M., Horváth L.: Limit Theorems in Change Point Analysis. Wiley, New York 1997 MR 2743035
[7] Embrechts P., Klüppelberg, C., Mikosch T.: Modelling Extremal Events. Springer–Verlag, Heildelberg 1997 MR 1458613 | Zbl 0873.62116
[8] Haccou P., Meelis, E., Geer S. van de: The likelihood ratio test for the change point problem for exponentially distributed random variables. Stochastic Process. Appl. 27 (1988), 121–139 MR 0934533
[9] Hájek J., Šidák Z.: Theory of Rank Tests. Academia, Prague 1967 MR 0229351 | Zbl 0944.62045
[10] Kander Z., Zacks S.: Test procedures for possible changes in parameters of statistical distributions occurring at unknown time points. Ann. Math. Statist. 37 (1966), 1196–1210 MR 0202242
[11] Kiefer J.: K-sample analogues of the Kolmogorov–Smirnov’s and Cramér–von Mises tests. Ann. Math. Statist. 30 (1960), 420–447 MR 0102882
[12] Kotz S., Balakrishnan, M., Johnson N. L.: Continuous Multivariate Distributions. Volume 1: Models and Applications. Wiley, New York 2000 MR 1788152 | Zbl 0946.62001
[13] Kvaløy J. T., Lindqvist B. H.: TTT-based tests for trend in repairable systems data. Reliability Engineering and System Safety 60 (1998), 13–28
[14] Kvaløy J. T., Lindqvist B. H., Malmedal H.: A statistical test for monotonic and non-monotonic trend in repairable systems. In: Proc. European Conference on Safety and Reliability – ESREL 2001, Torino 2002, pp. 1563–1570
[16] Sigma: Natural Catastrophes and Major Losef in 1995. Sigma Publ. 2 (1995)
[17] Steinebach. J., Eastwood V. R.: On extreme value asymptotics for increments of renewal processes. J. Statist. Plann. Inference 44 (1995) MR 1342102 | Zbl 0831.60060
[18] Steinebach J., Eastwood V. R.: Extreme value asymptotics for multivariate renewal processes. J. Multivariate Anal. 56 (1996), 284–302 MR 1379531 | Zbl 0861.60065
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