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

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. Neumann, Michael H., 2007. "Deconvolution from panel data with unknown error distribution," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1955-1968, November.
    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).
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