We observe inhomogeneous Poisson’s processes with covariates and aim at estimating their intensities. We assume that the intensity of each Poisson’s process is of the form where is a covariate and where is an unknown function. We propose a model selection approach where the models are used to approximate the multivariate function . We show that our estimator satisfies an oracle-type inequality under very weak assumptions both on the intensities and the models. By using an Hellinger-type loss, we establish non-asymptotic risk bounds and specify them under several kind of assumptions on the target function such as being smooth or a product function. Besides, we show that our estimation procedure is robust with respect to these assumptions. This procedure is of theoretical nature but yields results that cannot currently be obtained by more practical ones.
DOI : 10.1051/ps/2014022
Keywords: Adaptive estimation, model selection, Poisson processes, T-estimator
Sart, Mathieu 1
@article{PS_2015__19__204_0,
author = {Sart, Mathieu},
title = {Model selection for {Poisson} processes with covariates},
journal = {ESAIM: Probability and Statistics},
pages = {204--235},
year = {2015},
publisher = {EDP Sciences},
volume = {19},
doi = {10.1051/ps/2014022},
mrnumber = {3394490},
zbl = {1392.62253},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2014022/}
}
Sart, Mathieu. Model selection for Poisson processes with covariates. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 204-235. doi: 10.1051/ps/2014022
and , Wavelet shrinkage for natural exponential families with quadratic variance functions. Biometrika 88 (2001) 805–820. | MR | Zbl
, and , Wavelet shrinkage for natural exponential families with cubic variance functions. Sankhyā. Indian J. Stat., Series A 63 (2001) 309–327. | MR | Zbl
, Estimator selection with respect to Hellinger-type risks. Probab. Theory Relat. Fields 151 (2011) 353–401. | MR | Zbl
and , Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Relat. Fields 143 (2009) 239–284. | MR | Zbl
and , Estimating composite functions by model selection. Ann. Inst. Henri Poincaré, Probab. Stat. 50 (2014) 285–314. | MR | Zbl
and , Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 (1991) 1034–1054. | MR | Zbl
, Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré, Probab. Stat. 42 (2006) 273–325. | MR | Zbl
L. Birgé, Model selection for Poisson processes. In Asymptotics: particles, processes and inverse problems. In vol. 55 of IMS Lect. Notes Monogr. Ser. Inst. Math. Statist., Beachwood, OH (2007) 32–64. | MR | Zbl
L. Birgé, Model selection for density estimation with -loss. Probab. Theory Relat. Fields (2013) 1–42. | MR | Zbl
, and , Adaptive estimation of the conditional intensity of marker-dependent counting processes. Ann. Inst. Henri Poincaré, Probab. Stat. 47 (2011) 1171–1196. | MR | Zbl
, and , Multi-dimensional spline approximation. SIAM J. Numer. Anal. 17 (1980) 380–402. | MR | Zbl
, Learning curve approach to reliability monitoring. IEEE Trans. Aerospace 2 (1964) 563–566.
and , Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. R-28 (1979) 206–211. | Zbl
, and , Nonparametric estimation of composite functions. Ann. Stat. 37 (2009) 1360–1404. | MR | Zbl
, and , Multiscale photon-limited spectral image reconstruction. SIAM J. Imaging Sci. 3 (2010) 619–645. | MR | Zbl
P. Massart, Concentration inequalities and model selection. École d’été de Probabilités de Saint-Flour. In vol. 1896 of Lect. Notes Math. Springer, Berlin/Heidelberg (2003). | MR | Zbl
, Adaptive estimation of the intensity of inhomogeneous poisson processes via concentration inequalities. Probab. Theory Relat. Fields 126 (2003) 103–153. | MR | Zbl
, and , S-shaped reliability growth modeling for software error detection. IEEE Trans. Reliab. 32 (1983) 475–484.
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