Abstract
Under the presence of only one realization, we consider a computationally simple algorithm for estimating the intensity function of a Poisson process with exponential quadratic and cyclic of fixed frequency trends. We argue that the algorithm can successfully be used to estimate any Poisson intensity function provided that it has a parametric form.
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REFERENCES
Cox, D. R. (1972). The statistical analysis of dependencies in point processes, Stochastic Point Processes (ed. P. A. W. Lewis), 55–66, Wiley, New York.
Cox, D. R. and Isham, V. (1980). Point Processes, Chapman and Hall, London.
Cox, D. R. and Lewis, P. A. W. (1978). The Statistical Analysis of Series of Events, Chapman and Hall, London.
Cressie, N. A. C. (1991). Statistics for Spatial Data, Wiley, New York.
Daley, D. J. and Vere-Jones, D. (1972). A summary of the theory of point processes, Stochastic Point Processes (ed. P. A. W. Lewis), 299–383, Wiley, New York.
Daley, D. J. and Vere-Jones, D. (1988). Introduction to the Theory of Point Processes, Springer, New York.
Devroye, L. and Gyorfi, L. (1985). Nonparametric Density Estimation: The L1 View, Wiley, New York.
Diggle, P. J. (1983). Statistical Analysis of Spatial Point Processes, Academic Press, London.
Helmers, R. (1995). On estimating the intensity of oil-pollution in the North-Sea, Report BS-N9501. Centre for Mathematics and Computer Science, Amsterdam.
Kallenberg, O. (1983). Random Measures, Academic Press, New York.
Karr, A. F. (1986). Point Processes and Their Statistical Inference, Marcel Dekker, New York.
Kingman, J. F. C. (1993). Poisson Processes, Clarendon Press, Oxford.
Krickeberg, K. (1982). Processus ponctuels en statistique, Lecture Notes in Math. (ed. P. L. Hennequin), 929, 205–313, Springer, New York.
Kutoyants, Yu. A. (1984). Parameter Estimation for Stochastic Processes, Heldermann, Berlin.
Lewis, P. A. W. (1970). Remarks on the theory, computation and application of the spectral analysis of series of events, J. Sound Vibration, 12, 353–375.
Lewis, P. A. W. (1972). Recent results in the statistical analysis of univariate point processes, Stochastic Point Processes (ed. P. A. W. Lewis), 1 54, Wiley, New York.
Ogata, Y. and Katsura, K. (1986). Point process models with linearly parametrized intensity for application to earthquake data, J. Appl. Probab., 23A, 291–310.
Rathbun, S. L. and Crossie, N. (1994). Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes, Advances in Applied Probability, 26, 122–154.
Reiss, R.-D. (1993). A Course on Point Processes, Springer, New York.
Ripley, B. D. (1976). Locally finite random sets, foundations for point process theory, Ann. Probab., 4, 983–994.
Ripley, B. D. (1991). Statistical Inference for Spatial Processes, Cambridge University Press, Cambridge.
Rosenblatt, M. (1991). Stochastic Curve Estimation, NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 3, IMS, Hayward, and ASA, Alexandria.
Vere-Jones, D. (1995). Forecasting earthquakes and earthquake risk, International Journal of Forecasting, 11, 503–538.
Vere-Jones, D. and Ozaki, T. (1982). Some examples of statistical estimation applied to earthquake data, Ann. Inst. Statist. Math., 34, 189–207.
Zheng, X. and Vere-Jones, D. (1994). Further applications of the stochastic stress release model to historical earthquake data, Technophysics, 229, 101–121.
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Helmers, R., Zitikis, R. On Estimation of Poisson Intensity Functions. Annals of the Institute of Statistical Mathematics 51, 265–280 (1999). https://doi.org/10.1023/A:1003806107972
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DOI: https://doi.org/10.1023/A:1003806107972