IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v84y2003i1p19-39.html
   My bibliography  Save this article

Consistent estimation of the intensity function of a cyclic Poisson process

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(02)00008-8
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Ellis, Steven P., 1991. "Density estimation for point processes," Stochastic Processes and their Applications, Elsevier, vol. 39(2), pages 345-358, December.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    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. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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.
    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. Mat'uv{s} Maciak & Ostap Okhrin & Michal Pev{s}ta, 2019. "Infinitely Stochastic Micro Forecasting," Papers 1908.10636, arXiv.org, revised Sep 2019.
    5. 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.
    6. Dimitrios Tsitsis & George Karavasilis & Alexandros Rigas, 2012. "Measuring the association of stationary point processes using spectral analysis techniques," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(1), pages 23-47, March.
    7. 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.
    8. Youssef Taleb & Edward A. K. Cohen, 2021. "Multiresolution analysis of point processes and statistical thresholding for Haar wavelet-based intensity estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 395-423, April.
    9. Pierre Jacob & Paulo Oliveira, 1999. "Histograms and Associated Point Processes," Statistical Inference for Stochastic Processes, Springer, vol. 2(3), pages 227-251, October.
    10. Maciak, Matúš & Okhrin, Ostap & Pešta, Michal, 2021. "Infinitely stochastic micro reserving," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 30-58.
    11. José Miranda & Pedro Morettin, 2011. "Estimation of the intensity of non-homogeneous point processes via wavelets," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(6), pages 1221-1246, December.
    12. Florens, Danielle & Pham, Huyên, 1999. "Large deviation principle in nonparametric estimation of marked point processes," Statistics & Probability Letters, Elsevier, vol. 41(4), pages 383-388, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:84:y:2003:i:1:p:19-39. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.