Abstract
We consider the nonparametric estimation of the intensity function of a Poisson point process in a circular model from indirect observations \(N_1,\ldots ,N_n\). 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 \(Y_1,\ldots ,Y_m\) 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.
Similar content being viewed by others
References
Antoniadis A, Bigot J (2006) Poisson inverse problems. Ann Stat 34(5):2132–2158. https://doi.org/10.1214/009053606000000687
Baraud Y, Birgé L (2009) Estimating the intensity of a random measure by histogram type estimators. Probab Theory Relat Fields 143(1–2):239–284. https://doi.org/10.1007/s00440-007-0126-6
Barron A, Birgé L, Massart P (1999) Risk bounds for model selection via penalization. Probab Theory Relat Fields 113(3):301–413. https://doi.org/10.1007/s004400050210
Bigot J, Gadat S, Klein T, Marteau C (2013) Intensity estimation of non-homogeneous Poisson processes from shifted trajectories. Electron J Stat 7:881–931. https://doi.org/10.1214/13-EJS794
Birgé L (2007) Model selection for Poisson processes. In: Asymptotics: particles, processes and inverse problems, IMS Lecture Notes Monograph Series, vol 55. Institute of Mathematical Statistics, Beachwood, pp 32–64. https://doi.org/10.1214/074921707000000265
Brémaud P (1981) Point processes and queues. Martingale dynamics. Springer Series in Statistics. Springer, New York, Berlin
Cavalier L, Koo JY (2002) Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inform. Theory 48(10):2794–2802. https://doi.org/10.1109/TIT.2002.802632
Chiu SN, Stoyan D, Kendall WS, Mecke J (2013) Stochastic geometry and its applications, 3rd edn. Wiley series in probability and statistics. Wiley, Chichester. https://doi.org/10.1002/9781118658222
Comte F (2015) Estimation non-paramétrique. Spartacus, Paris
Comte F, Lacour C (2010) Pointwise deconvolution with unknown error distribution. C R Math Acad Sci Paris 348(5–6):323–326. https://doi.org/10.1016/j.crma.2010.02.012
Comte F, Lacour C (2011) Data-driven density estimation in the presence of additive noise with unknown distribution. J R Stat Soc Ser B Stat Methodol 73(4):601–627. https://doi.org/10.1111/j.1467-9868.2011.00775.x
Delattre S, Hoffmann M, Picard D, Vareschi T (2012) Blockwise SVD with error in the operator and application to blind deconvolution. Electron J Stat 6:2274–2308. https://doi.org/10.1214/12-EJS745
Diggle PJ, Hall P (1993) A Fourier approach to nonparametric deconvolution of a density estimate. J R Stat Soc Ser B 55(2):523–531
Fan J (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Ann Stat 19(3):1257–1272. https://doi.org/10.1214/aos/1176348248
Grégoire G, Nembé J (2000) Convergence rates for the minimum complexity estimator of counting process intensities. J Nonparametr Stat 12(5):611–643. https://doi.org/10.1080/10485250008832825
Johannes J (2009) Deconvolution with unknown error distribution. Ann Stat 37(5A):2301–2323. https://doi.org/10.1214/08-AOS652
Johannes J, Schwarz M (2013a) Adaptive circular deconvolution by model selection under unknown error distribution. Bernoulli 19(5A):1576–1611. https://doi.org/10.3150/12-BEJ422
Johannes J, Schwarz M (2013b) Adaptive Gaussian inverse regression with partially unknown operator. Commun Stat Theory Methods 42(7):1343–1362. https://doi.org/10.1080/03610926.2012.731548
Karr AF (1991) Point processes and their statistical inference, probability: pure and applied, vol 7, 2nd edn. Marcel Dekker Inc, New York
Kroll, M (2016) Concentration inequalities for Poisson point processes with application to adaptive intensity estimation. arXiv:1612.07901
Kutoyants YA (1998) Statistical inference for spatial Poisson processes, vol 134. Lecture Notes in Statistics. Springer, New York
Lacour C (2006) Rates of convergence for nonparametric deconvolution. C R Math Acad Sci Paris 342(11):877–882. https://doi.org/10.1016/j.crma.2006.04.006
Massart P (2007) Concentration inequalities and model selection, vol 1896. Lecture Notes in Mathematics. Springer, Berlin
Meister A (2009) Deconvolution problems in nonparametric statistics, vol 193. Lecture Notes in Statistics. Springer, Berlin
Mitzenmacher M, Upfal E (2017) Probability and computing, 2nd edn. Cambridge University Press, Cambridge Randomization and probabilistic techniques in algorithms and data analysis
Neumann MH (2007) Deconvolution from panel data with unknown error distribution. J Multivar Anal 98(10):1955–1968. https://doi.org/10.1016/j.jmva.2006.09.012
Patil PN, Wood ATA (2004) Counting process intensity estimation by orthogonal wavelet methods. Bernoulli 10(1):1–24. https://doi.org/10.3150/bj/1077544601
Petrov VV (1995) Limit theorems of probability theory, vol 4. Oxford studies in probability. Sequences of independent random variables. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York
Reiss RD (1989) Approximate distributions of order statistics. Springer series in statistics. With applications to nonparametric statistics. Springer, New York. https://doi.org/10.1007/978-1-4613-9620-8
Reiss RD (1993) A course on point processes. Springer series in statistics. Springer, New York. https://doi.org/10.1007/978-1-4613-9308-5
Resnick SI (2008) Extreme values, regular variation and point processes. Springer series in operations research and financial engineering. Springer, New York (Reprint of the 1987 original)
Reynaud-Bouret P (2003) Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab Theory Relat Fields 126(1):103–153. https://doi.org/10.1007/s00440-003-0259-1
Sansonnet L (2014) Wavelet thresholding estimation in a Poissonian interactions model with application to genomic data. Scand J Stat 41(1):200–226. https://doi.org/10.1111/sjos.12009
Schwarz M, Van Bellegem S (2010) Consistent density deconvolution under partially known error distribution. Stat Probab Lett 80(3–4):236–241. https://doi.org/10.1016/j.spl.2009.10.012
Serfozo R (2009) Basics of applied stochastic processes. Probability and its applications (New York). Springer, Berlin. https://doi.org/10.1007/978-3-540-89332-5
Shen JJ, Zhang NR (2012) Change-point model on nonhomogeneous Poisson processes with application in copy number profiling by next-generation DNA sequencing. Ann Appl Stat 6(2):476–496. https://doi.org/10.1214/11-AOAS517
Tsybakov AB (2009) Introduction to nonparametric estimation. Springer series in statistics. Springer, New York. https://doi.org/10.1007/b13794 (Revised and extended from the 2004 French original, Translated by Vladimir Zaiats)
Zhang T, Kou SC (2010) Nonparametric inference of doubly stochastic Poisson process data via the kernel method. Ann Appl Stat 4(4):1913–1941. https://doi.org/10.1214/10-AOAS352
Acknowledgements
Financial support by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group RTG 1953 is gratefully acknowledged. I am indepted to my supervisors Jan Johannes and Martin Schlather for fruitful discussions and helpful comments on the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kroll, M. Nonparametric intensity estimation from noisy observations of a Poisson process under unknown error distribution. Metrika 82, 961–990 (2019). https://doi.org/10.1007/s00184-019-00716-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00716-7