Nonlinear estimation problems of poisson cluster processes

  • Franz Konecny
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 84)


Doubly stochastic Poisson processes whose unobservable intensity is a shot-noise process with random amplitude arise when each event of a primary Poisson process generates a random number of subsidiary events. We derive a stochastic partial differential equation for the unnormalized conditional moment generating function. This equation can be used for recursive compution of the minimum variance estimator of the unobservable intensity as well as the likelihood ratio with respect to the reference measure, on the basis of point process observations.


Poisson Process Point Process Stochastic Partial Differential Equation Random Amplitude Stochastic Intensity 
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  1. [1]
    Bartlett, M.S. (1964). The spectral analysis of two-dimensional point processes. Biometrika 51, 299–311.Google Scholar
  2. [2]
    Boel, R.K. and Benes, V.E. (1980). Recursive Nonlinear Estimation of a Diffusion Acting as the Rate of an Observed Poisson Process. IEEE Trans.Inform.Theory IT-26/5, 561–575.Google Scholar
  3. [3]
    Bremáud, P. (1981). Point Processes and Queues, Martingale Dynamics. Springer-Verlag, Berlin.Google Scholar
  4. [4]
    Lawrence, A.J. (1972). Some models for stationary series of univariate events. In: P.A.W. Lewis (ed.), Stochastic Point Processes, Wiley, New York.Google Scholar
  5. [5]
    Kavvas, M.L. and Delleur, J.W. (1981). A Stochastic Cluster Model of Daily Rainfall Sequences. Water Resour.Res. 17/4, 1151–1160.Google Scholar
  6. [6]
    v. Schuppen, J.H. (1977). Filtering, prediction and smoothing for counting process observations, a martingale approach. SIAM J.appl.Math 32/3.Google Scholar
  7. [7]
    Smith, J.A. and Karr A.F. (1984). Statistical inference for point process models of rainfall. Water Resour.Res., to appear.Google Scholar
  8. [8]
    Snyder, D.L. (1975). Random Point Processes. Wiley, New York.Google Scholar
  9. [9]
    Vere-Jones, D. (1970). Stochastic models for earthquake occurrence (with discussion). J.R. Statist.Soc. 8 32, 1–62.Google Scholar
  10. [10]
    Ogata, Y. and Akaike, H. (1982). On the Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-exciting Point Processes. J.R. Statist.Soc.B 44/1, 102–107.Google Scholar
  11. [11]
    Konecny, F. (1985). On the estimation of the stochastic intensity and the parameters of Neyman-Scott trigger processes (submitted to Mathem. Operationsfoschung und Statistik).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Franz Konecny
    • 1
  1. 1.Institut für Mathematik u. Angewandte StatistikUniversität für Bodenkultur - WienWienAustria

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