Consistent estimation of the intensity function of a cyclic Poisson process

Journal of Multivariate Analysis

Volumn 84, Issue 1, 2003, Pages 19-39

  Helmers, Roelof ^a   Wayan Mangku, I ^b   Zitikis, Ričardas ^c  

^a CWI   (Netherlands)

^b BOGOR AGRICULTURAL UNIVERSITY   (Indonesia)

^c UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Consistency; Cyclic intensity function; Cyclic Poisson process; Nonparametric estimation; Periodic intensity function; Periodic Poisson process; Point process; Poisson process; Rate of consistency

Indexed keywords

EID: 0037288739     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0047-259X(02)00008-8     Document Type: Review

References (54)

1. O. Aalen, Nonparametric inference for a family of counting processes, Ann. Statist. 6 (1978) 701-726.
2. F. Ashkar, J. Rousselle, 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 (1983) 481-485.
3. D.T. Chen, M. Rieders, Cyclic modulated Poisson processes in traffic characterization, Comm. Statist. Stochastic Models 12 (1996) 585-610.
4. S. Chukova, B. Dimitrov, J. Garrido, Renewal and nonhomogeneous Poisson processes generated by distribution with periodic failure rate, Statist. Probab. Lett. 17 (1993) 19-25.
5. A. Cowling, P. Hall, M.J. Phillips, Bootstrap confidence regions for the intensity of a Poisson point process, J. Amer. Statist. Assoc. 91 (1996) 1516-1524.
6. D.R. Cox, V. Isham, Point Processes, Chapman & Hall, London, 1980.
7. D.R. Cox, P.A.W. Lewis, The Statistical Analysis of Series of Events, Chapman & Hall, London, 1978.
8. N.A.C. Cressie, Statistics for Spatial Data, Wiley, New York, 1991.
9. D.J. Daley, D. Vere-Jones, Introduction to the Theory of Point Processes, Springer, New York, 1988.
10. G. Dia, Nonparametric estimation of the density of a point process, Statist. Probab. Lett. 10 (1990) 397-405.
11. P.J. Diggle, Statistical Analysis of Spatial Point Processes, Academic Press, London, 1983.
12. P. Diggle, J.S. Marron, Equivalence of smoothing parameter selectors in density and intensity estimation, J. Amer. Statist. Assoc. 83 (1988) 793-800.
13. B. Dimitrov, S. Chukova, Environmental modeling in driving periodic conditions, in: B.I. Cheshankov, M.D. Todorov (Eds.), Applications of Mathematics in Engineering, Heron Press, Sofia, 1999, pp. 21-30.
14. B. Dimitrov, S. Chukova, M. El-Saidi, Modeling uncertainty in periodic random environment: applications to environmental studies, Environmetrics 10 (1999) 467-485.
15. B. Dimitrov, S. Chukova, D. Green, Probability distributions in periodic random environment and their applications, SIAM J. Appl. Math. 57 (1997) 501-517.
16. S.P. Ellis, Density estimation for point processes, Stochastic Process. Appl. 39 (1991) 345-358.
17. J.S. Forrest, Variations in thunderstorm severity in Great Britain, Quart. J. Roy. Meteorol. Soc. 76 (1950) 277-286.
18. R.M. Gagliardi, S. Karp, Optical Communications, Wiley, New York, 1976.
19. L. Guenni, F. Ojeda, M.C. Key, Periodic model selection for rainfall using conditional maximum likelihood, Environmetrics 9 (1998) 407-417.
20. R. Helmers, R. Zitikis, On estimation of Poisson intensity functions, Ann. Inst. Statist. Math. 51 (1999) 265-280.
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.
22. E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.
23. A.F. Karr, Point Processes and their Statistical Inference, Dekker, New York, 1986.
24. J.F.C. Kingman, Poisson Processes, Clarendon, Oxford, 1993.
25. F. Konecny, The asymptotic properties of maximum likelihood estimators for marked Poisson processes with a cyclic intensity measure, Metrika 34 (1987) 143-155.
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.
27. Yu.A. Kutoyants, Statistical Inference for Spatial Poisson Processes, in: Lecture Notes in Statistics, Vol. 134, Springer, New York, 1998.
28. S. Lee, J.R. Wilson, M.M. Crawford, Modeling and simulation of a nonhomogeneous Poisson process having cyclic behavior, Comm. Statist. Simulation 20 (1991) 777-809.
29. P.A.W. Lewis, Remarks on the theory, computation and application of the spectral analysis of series of events, J. Sound Vib. 12 (1970) 353-375.
30. P.A.W. Lewis, Recent results in the statistical analysis of univariate point processes, in: P.A.W. Lewis (Ed.), Stochastic Point Processes, Wiley, New York, 1972, pp. 1-54.
31. P.A.W. Lewis (Ed.), Stochastic Point Processes, Wiley, New York, 1972.
32. L. Mandel, Fluctuations in photon beams and their correlations, Proc. Phys. Soc. 72 (1958) 1037-1048.
33. L. Mandel, Fluctuations in photon beams: the distribution of the photon-electrons, Proc. Phys. Soc. 74 (1959) 233-243.
34. I.W. Mangku, Estimating the intensity of a cyclic Poisson process, Ph.D. Thesis, University of Amsterdam, Amsterdam, 2001.
35. K. Matsumura, On regional characteristics of seasonal variation of shallow earthquake activities in the word, Bull. Disas. Prev. Res. Inst. Kyoto Univ. 36 (1986) 43-98.
36. Y. Ogata, Likelihood analysis of point processes and its application to seismological data, Bull. Int. Statist. Inst. 50 (1983) 943-961.
37. Y. Ogata, Seismicity analysis through point-process modeling: a review, Pure Appl. Geophys. 155 (1999) 471-507.
38. Y. Ogata, K. Katsura, Point-process models with linearly parameterized intensity for application to earthquake data, J. Appl. Probab. A 23 (1986) 291-310.
39. O. Pons, Vitesse de convergence des estimateurs à noyau pour l'intensité d'un processus ponctuel, Statistics 17 (1986) 577-584.
40. H. Ramlau-Hansen, Smoothing counting process intensities by means of kernel functions, Ann. Statist. 11 (1983) 453-466.
41. S.L. Rathbun, N. Cressie, Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes, Adv. Appl. Probab. 26 (1994) 122-154.
42. R.-D. Reiss, A Course on Point Processes, Springer, New York, 1993.
43. W.M. Siebert, Stimulus transformations in the peripheral auditory system, in: P.A. Kolers, M. Eden (Eds.), Recognizing Patterns: Studies in Living and Automated Systems, MIT Press, Cambridge, MA, 1968, pp. 104-133.
44. W.M. Siebert, Frequency discrimination in the auditory system: Place or periodicity mechanism, Proc. IEEE 58 (1970) 723-730.
45. J.A. Smith, A.F. Karr, A point process model of summer season rainfall occurrences, Water Resources Res. 19 (1983) 95-103.
46. D.L. Snyder, M.I. Miller, Random Point Processes in Time and Space, 2nd Edition, Springer, New York, 1995.
47. P. Todorovic, E. Zelenhasic, A stochastic model for flood analysis, Water Resources Res. 6 (1970) 1641-1648.
48. D. Vere-Jones, Stochastic models for earthquake occurrence (with discussion), J. R. Statist. Soc. Ser. B 32 (1970) 1-62.
49. D. Vere-Jones, On the estimation of frequency in point-process data, J. Appl. Probab. A 19 (1982) 383-394.
50. D. Vere-Jones, Statistical Seismology, Lecture Notes, Academia Sinica University of Science and Technology of China, 1995.
51. D. Vere-Jones, T. Ozaki, Some examples of statistical estimation applied to earthquake data, Ann. Inst. Statist. Math. B 34 (1982) 189-207.
52. E. Waymire, V.K. Gupta, The mathematical structure of rainfall representations, 1: a review of the stochastic rainfall models, Water Resources Res. 17 (1981) 1261-1272.
53. E. Waymire, V.K. Gupta, The mathematical structure of rainfall representations, 2: a review of the theory of point processes, Water Resources Res. 17 (1981) 1273-1285.
54. E. Waymire, V.K. Gupta, The mathematical structure of rainfall representations, 3: some applications of the point process theory to rainfall processes, Water Resources Res. 17 (1981) 1287-1294.