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Consistent estimation of the intensity function of a cyclic Poisson process

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  • Helmers, Roelof
  • Wayan Mangku, I.
  • Zitikis, Ricardas

Abstract

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.

Suggested Citation

  • Helmers, Roelof & Wayan Mangku, I. & Zitikis, Ricardas, 2003. "Consistent estimation of the intensity function of a cyclic Poisson process," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 19-39, January.
    • Handle: RePEc:eee:jmvana:v:84:y:2003:i:1:p:19-39
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    References listed on IDEAS

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    1. Ellis, Steven P., 1991. "Density estimation for point processes," Stochastic Processes and their Applications, Elsevier, vol. 39(2), pages 345-358, December.
    2. Roelof Helmers & Ričardas Zitikis, 1999. "On Estimation of Poisson Intensity Functions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(2), pages 265-280, June.
    3. Franz Konecny, 1987. "The asymptotic properties of maximum likelihood estimators for marked poisson processes with a cyclic intensity measure," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 34(1), pages 143-155, December.
    4. Chukova, Stefanka & Dimitrov, Boyan & Garrido, José, 1993. "Renewal and nonhomogeneous Poisson processes generated by distributions with periodic failure rate," Statistics & Probability Letters, Elsevier, vol. 17(1), pages 19-25, May.
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    Cited by:

    1. Mark Bebbington & Ričardas Zitikis, 2004. "A Robust Heuristic Estimator for the Period of a Poisson Intensity Function," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 441-462, December.
    2. Nan Shao & Keh‐Shin Lii, 2011. "Modelling non‐homogeneous Poisson processes with almost periodic intensity functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 99-122, January.
    3. Roelof Helmers & I. Mangku, 2009. "Estimating the intensity of a cyclic Poisson process in the presence of linear trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 599-628, September.
    4. Roelof Helmers & I. Mangku, 2012. "Predicting a cyclic Poisson process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1261-1279, December.
    5. Roelof Helmers & Qiying Wang & Ričardas Zitikis, 2009. "Confidence regions for the intensity function of a cyclic Poisson process," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 21-36, February.
    6. Froda, Sorana & Ferland, René, 2012. "Estimating the parameters of a Poisson process model for predator–prey interactions," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2252-2259.
    7. Helmers, Roelof & Mangku, I. Wayan & Zitikis, Ricardas, 2005. "Statistical properties of a kernel-type estimator of the intensity function of a cyclic Poisson process," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 1-23, January.
    8. Camerlenghi, F. & Capasso, V. & Villa, E., 2014. "On the estimation of the mean density of random closed sets," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 65-88.

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