Consistent estimation of the intensity function of a cyclic Poisson process. (English) Zbl 1038.62037
Summary: We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We do not assume any particular parametric form for the intensity function, nor do we even assume that it is continuous. Moreover, we consider the situation when only a single realization of the Poisson process is available, and only in a bounded window. We prove, in particular, that the proposed estimator is consistent when the size of the window indefinitely expands. We also obtain complete convergence of the estimator.
Keywords:
Poisson process; Cyclic Poisson process; Periodic Poisson process; Point process; Cyclic intensity function; Periodic intensity function; Nonparametric estimation; Consistency; Rate of consistencyReferences:
| [1] | Aalen, O., Nonparametric inference for a family of counting processes, Ann. Statist., 6, 701-726 (1978) · Zbl 0389.62025 |
| [2] | Ashkar, F.; Rousselle, J., The effect of certain restrictions imposed on the inter-arrival times of flood events on the Poisson distribution used for modeling flood counts, Water Resources Res., 19, 481-485 (1983) |
| [3] | Chen, D. T.; Rieders, M., Cyclic modulated Poisson processes in traffic characterization, Comm. Statist. Stochastic Models, 12, 585-610 (1996) · Zbl 0873.60071 |
| [4] | Chukova, S.; Dimitrov, B.; Garrido, J., Renewal and nonhomogeneous Poisson processes generated by distribution with periodic failure rate, Statist. Probab. Lett., 17, 19-25 (1993) · Zbl 0771.60065 |
| [5] | Cowling, A.; Hall, P.; Phillips, M. J., Bootstrap confidence regions for the intensity of a Poisson point process, J. Amer. Statist. Assoc., 91, 1516-1524 (1996) · Zbl 0882.62078 |
| [6] | Cox, D. R.; Isham, V., Point Processes (1980), Chapman & Hall: Chapman & Hall London · Zbl 0441.60053 |
| [7] | Cox, D. R.; Lewis, P. A.W., The Statistical Analysis of Series of Events (1978), Chapman & Hall: Chapman & Hall London · Zbl 0148.14005 |
| [8] | Cressie, N. A.C., Statistics for Spatial Data (1991), Wiley: Wiley New York · Zbl 0468.62095 |
| [9] | Daley, D. J.; Vere-Jones, D., Introduction to the Theory of Point Processes (1988), Springer: Springer New York · Zbl 0657.60069 |
| [10] | Dia, G., Nonparametric estimation of the density of a point process, Statist. Probab. Lett., 10, 397-405 (1990) · Zbl 0724.62041 |
| [11] | Diggle, P. J., Statistical Analysis of Spatial Point Processes (1983), Academic Press: Academic Press London · Zbl 0559.62088 |
| [12] | Diggle, P.; Marron, J. S., Equivalence of smoothing parameter selectors in density and intensity estimation, J. Amer. Statist. Assoc., 83, 793-800 (1988) · Zbl 0662.62036 |
| [13] | Dimitrov, B.; Chukova, S., Environmental modeling in driving periodic conditions, (Cheshankov, B. I.; Todorov, M. D., Applications of Mathematics in Engineering (1999), Heron Press: Heron Press Sofia), 21-30 |
| [14] | Dimitrov, B.; Chukova, S.; El-Saidi, M., Modeling uncertainty in periodic random environmentapplications to environmental studies, Environmetrics, 10, 467-485 (1999) |
| [15] | Dimitrov, B.; Chukova, S.; Green, D., Probability distributions in periodic random environment and their applications, SIAM J. Appl. Math., 57, 501-517 (1997) · Zbl 0869.60008 |
| [16] | Ellis, S. P., Density estimation for point processes, Stochastic Process. Appl., 39, 345-358 (1991) · Zbl 0749.62023 |
| [17] | Forrest, J. S., Variations in thunderstorm severity in Great Britain, Quart. J. Roy. Meteorol. Soc., 76, 277-286 (1950) |
| [18] | Gagliardi, R. M.; Karp, S., Optical Communications (1976), Wiley: Wiley New York |
| [19] | Guenni, L.; Ojeda, F.; Key, M. C., Periodic model selection for rainfall using conditional maximum likelihood, Environmetrics, 9, 407-417 (1998) |
| [20] | Helmers, R.; Zitikis, R., On estimation of Poisson intensity functions, Ann. Inst. Statist. Math., 51, 265-280 (1999) · Zbl 0946.62076 |
| [21] | C. Helstrom, Estimation of modulation frequency of a light beam, in: R.S. Kennedy, S. Karp (Eds.), Optical Space Communication, Appendix E, Proceedings of a Workshop Held at Williams College, Williamstown, MA, 1968.; C. Helstrom, Estimation of modulation frequency of a light beam, in: R.S. Kennedy, S. Karp (Eds.), Optical Space Communication, Appendix E, Proceedings of a Workshop Held at Williams College, Williamstown, MA, 1968. |
| [22] | Hewitt, E.; Stromberg, K., Real and Abstract Analysis (1965), Springer: Springer New York · Zbl 0137.03202 |
| [23] | Karr, A. F., Point Processes and their Statistical Inference (1986), Dekker: Dekker New York · Zbl 0601.62120 |
| [24] | Kingman, J. F.C., Poisson Processes (1993), Clarendon: Clarendon Oxford · Zbl 0771.60001 |
| [25] | Konecny, F., The asymptotic properties of maximum likelihood estimators for marked Poisson processes with a cyclic intensity measure, Metrika, 34, 143-155 (1987) · Zbl 0629.62089 |
| [26] | K. Krickeberg, Processus ponctuels en statistique, in: P.L. Hennequin (Ed.), Lecture Notes in Mathematics, Vol. 929, Springer, New York, 1982, pp. 205-313.; K. Krickeberg, Processus ponctuels en statistique, in: P.L. Hennequin (Ed.), Lecture Notes in Mathematics, Vol. 929, Springer, New York, 1982, pp. 205-313. · Zbl 0486.62089 |
| [27] | Kutoyants, Yu. A., Statistical Inference for Spatial Poisson Processes, Lecture Notes in Statistics, Vol. 134 (1998), Springer: Springer New York · Zbl 0904.62108 |
| [28] | Lee, S.; Wilson, J. R.; Crawford, M. M., Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior, Comm. Statist. Simulation, 20, 777-809 (1991) · Zbl 0825.65123 |
| [29] | Lewis, P. A.W., Remarks on the theory, computation and application of the spectral analysis of series of events, J. Sound Vib., 12, 353-375 (1970) · Zbl 0196.21502 |
| [30] | Lewis, P. A.W., Recent results in the statistical analysis of univariate point processes, (Lewis, P. A.W., Stochastic Point Processes (1972), Wiley: Wiley New York), 1-54 · Zbl 0263.62056 |
| [31] | P.A.W. Lewis (Ed.), Stochastic Point Processes, Wiley, New York, 1972.; P.A.W. Lewis (Ed.), Stochastic Point Processes, Wiley, New York, 1972. · Zbl 0238.00014 |
| [32] | Mandel, L., Fluctuations in photon beams and their correlations, Proc. Phys. Soc., 72, 1037-1048 (1958) |
| [33] | Mandel, L., Fluctuations in photon beamsthe distribution of the photon-electrons, Proc. Phys. Soc., 74, 233-243 (1959) |
| [34] | I.W. Mangku, Estimating the intensity of a cyclic Poisson process, Ph.D. Thesis, University of Amsterdam, Amsterdam, 2001.; I.W. Mangku, Estimating the intensity of a cyclic Poisson process, Ph.D. Thesis, University of Amsterdam, Amsterdam, 2001. |
| [35] | Matsumura, K., On regional characteristics of seasonal variation of shallow earthquake activities in the word, Bull. Disas. Prev. Res. Inst. Kyoto Univ., 36, 43-98 (1986) |
| [36] | Ogata, Y., Likelihood analysis of point processes and its application to seismological data, Bull. Int. Statist. Inst., 50, 943-961 (1983) |
| [37] | Ogata, Y., Seismicity analysis through point-process modelinga review, Pure Appl. Geophys., 155, 471-507 (1999) |
| [38] | Ogata, Y.; Katsura, K., Point-process models with linearly parameterized intensity for application to earthquake data, J. Appl. Probab. A, 23, 291-310 (1986) · Zbl 0583.62100 |
| [39] | Pons, O., Vitesse de convergence des estimateurs à noyau pour l’intensité d’un processus ponctuel, Statistics, 17, 577-584 (1986) · Zbl 0616.62049 |
| [40] | Ramlau-Hansen, H., Smoothing counting process intensities by means of kernel functions, Ann. Statist., 11, 453-466 (1983) · Zbl 0514.62050 |
| [41] | Rathbun, S. L.; Cressie, N., Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes, Adv. Appl. Probab., 26, 122-154 (1994) · Zbl 0811.62090 |
| [42] | Reiss, R.-D., A Course on Point Processes (1993), Springer: Springer New York · Zbl 0771.60037 |
| [43] | Siebert, W. M., Stimulus transformations in the peripheral auditory system, (Kolers, P. A.; Eden, M., Recognizing Patterns: Studies in Living and Automated Systems (1968), MIT Press: MIT Press Cambridge, MA), 104-133 |
| [44] | Siebert, W. M., Frequency discrimination in the auditory systemPlace or periodicity mechanism, Proc. IEEE, 58, 723-730 (1970) |
| [45] | Smith, J. A.; Karr, A. F., A point process model of summer season rainfall occurrences, Water Resources Res., 19, 95-103 (1983) |
| [46] | Snyder, D. L.; Miller, M. I., Random Point Processes in Time and Space (1995), Springer: Springer New York |
| [47] | Todorovic, P.; Zelenhasic, E., A stochastic model for flood analysis, Water Resources Res., 6, 1641-1648 (1970) |
| [48] | Vere-Jones, D., Stochastic models for earthquake occurrence, J. R. Statist. Soc. Ser. B, 32, 1-62 (1970), (with discussion) · Zbl 0201.27702 |
| [49] | Vere-Jones, D., On the estimation of frequency in point-process data, J. Appl. Probab. A, 19, 383-394 (1982) · Zbl 0516.62088 |
| [50] | D. Vere-Jones, Statistical Seismology, Lecture Notes, Academia Sinica University of Science and Technology of China, 1995.; D. Vere-Jones, Statistical Seismology, Lecture Notes, Academia Sinica University of Science and Technology of China, 1995. |
| [51] | Vere-Jones, D.; Ozaki, T., Some examples of statistical estimation applied to earthquake data, Ann. Inst. Statist. Math. B, 34, 189-207 (1982) |
| [52] | Waymire, E.; Gupta, V. K., The mathematical structure of rainfall representations, 1a review of the stochastic rainfall models, Water Resources Res., 17, 1261-1272 (1981) |
| [53] | Waymire, E.; Gupta, V. K., The mathematical structure of rainfall representations, 2a review of the theory of point processes, Water Resources Res., 17, 1273-1285 (1981) |
| [54] | Waymire, E.; Gupta, V. K., The mathematical structure of rainfall representations, 3some applications of the point process theory to rainfall processes, Water Resources Res., 17, 1287-1294 (1981) |
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