IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v82y2019i8d10.1007_s00184-019-00716-7.html
   My bibliography  Save this article

Nonparametric intensity estimation from noisy observations of a Poisson process under unknown error distribution

Author

Listed:
  • Martin Kroll

    (Universität Mannheim
    ENSAE ParisTech-CREST)

Abstract

We consider the nonparametric estimation of the intensity function of a Poisson point process in a circular model from indirect observations $$N_1,\ldots ,N_n$$ N 1 , … , N n . These observations emerge from hidden point process realizations with the target intensity through contamination with additive error. In case that the error distribution can only be estimated from an additional sample $$Y_1,\ldots ,Y_m$$ Y 1 , … , Y m we derive minimax rates of convergence with respect to the sample sizes n and m under abstract smoothness conditions and propose an orthonormal series estimator which attains the optimal rate of convergence. The performance of the estimator depends on the correct specification of a dimension parameter whose optimal choice relies on smoothness characteristics of both the intensity and the error density. We propose a data-driven choice of the dimension parameter based on model selection and show that the adaptive estimator attains the minimax optimal rate.

Suggested Citation

  • Martin Kroll, 2019. "Nonparametric intensity estimation from noisy observations of a Poisson process under unknown error distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(8), pages 961-990, November.
  • Handle: RePEc:spr:metrik:v:82:y:2019:i:8:d:10.1007_s00184-019-00716-7
    DOI: 10.1007/s00184-019-00716-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00184-019-00716-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s00184-019-00716-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Laure Sansonnet, 2014. "Wavelet Thresholding Estimation in a Poissonian Interactions Model with Application to Genomic Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 200-226, March.
    2. F. Comte & C. Lacour, 2011. "Data‐driven density estimation in the presence of additive noise with unknown distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 601-627, September.
    3. Neumann, Michael H., 2007. "Deconvolution from panel data with unknown error distribution," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1955-1968, November.
    4. Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Consistent density deconvolution under partially known error distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 236-241, February.
    5. Schwarz, M. & Van Bellegem, S., 2010. "Consistent density deconvolution under partially known error distribution," LIDAM Reprints ISBA 2010013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    Full references (including those not matched with items on IDEAS)

    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. Florens, Jean-Pierre & Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Nonparametric Frontier Estimation from Noisy Data," IDEI Working Papers 625, Institut d'Économie Industrielle (IDEI), Toulouse.
    2. Fabienne Comte & Adeline Samson, 2012. "Nonparametric estimation of random-effects densities in linear mixed-effects model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 951-975, December.
    3. Jochmans, Koen & Henry, Marc & Salanié, Bernard, 2017. "Inference On Two-Component Mixtures Under Tail Restrictions," Econometric Theory, Cambridge University Press, vol. 33(3), pages 610-635, June.
    4. Jun Cai & William C. Horrace & Christopher F. Parmeter, 2021. "Density deconvolution with Laplace errors and unknown variance," Journal of Productivity Analysis, Springer, vol. 56(2), pages 103-113, December.
    5. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    6. Kneip, Alois & Simar, Léopold & Van Keilegom, Ingrid, 2015. "Frontier estimation in the presence of measurement error with unknown variance," Journal of Econometrics, Elsevier, vol. 184(2), pages 379-393.
    7. Jean-Pierre Florens & Léopold Simar & Ingrid Van Keilegom, 2020. "Estimation of the Boundary of a Variable Observed With Symmetric Error," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 425-441, January.
    8. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2020. "Robust frontier estimation from noisy data: A Tikhonov regularization approach," Econometrics and Statistics, Elsevier, vol. 14(C), pages 1-23.
    9. D’Haultfœuille, Xavier & Février, Philippe, 2015. "Identification of mixture models using support variations," Journal of Econometrics, Elsevier, vol. 189(1), pages 70-82.
    10. Zhuan Pei & Yi Shen, 2017. "The Devil is in the Tails: Regression Discontinuity Design with Measurement Error in the Assignment Variable," Advances in Econometrics, in: Regression Discontinuity Designs, volume 38, pages 455-502, Emerald Group Publishing Limited.
    11. Johanna Kappus & Gwennaelle Mabon, 2013. "Adaptive Density Estimation in Deconvolution Problems with Unknown Error Distribution," Working Papers 2013-31, Center for Research in Economics and Statistics.
    12. Jeon, Jeong Min & Van Keilegom, Ingrid, 2023. "Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    13. Aurélie Bertrand & Ingrid Van Keilegom & Catherine Legrand, 2019. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," Biometrics, The International Biometric Society, vol. 75(1), pages 297-307, March.
    14. Gwennaëlle Mabon, 2014. "Adaptive Estimation of Random-Effects Densities In Linear Mixed-Effects Model," Working Papers 2014-41, Center for Research in Economics and Statistics.
    15. Bertrand, Aurelie & Van Keilegom, Ingrid & Legrand, Catherine, 2017. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," LIDAM Discussion Papers ISBA 2017025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Christophe Chesneau & Fabienne Comte & Gwennaëlle Mabon & Fabien Navarro, 2014. "Estimation of Convolution In The Model with Noise," Working Papers 2014-39, Center for Research in Economics and Statistics.
    17. Katerina Papagiannouli, 2022. "A Lepskiĭ-type stopping rule for the covariance estimation of multi-dimensional Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 25(3), pages 505-535, October.
    18. Van Ha Hoang & Thanh Mai Pham Ngoc & Vincent Rivoirard & Viet Chi Tran, 2022. "Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 4-43, March.
    19. Carrasco, Marine & Florens, Jean-Pierre, 2011. "A Spectral Method For Deconvolving A Density," Econometric Theory, Cambridge University Press, vol. 27(3), pages 546-581, June.
    20. Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Consistent density deconvolution under partially known error distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 236-241, 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:spr:metrik:v:82:y:2019:i:8:d:10.1007_s00184-019-00716-7. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.