Adaptive tests of homogeneity for a Poisson process

M. Fromont; B. Laurent; P. Reynaud-Bouret

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 1, page 176-213
  • ISSN: 0246-0203

Abstract

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We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in -norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.

How to cite

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Fromont, M., Laurent, B., and Reynaud-Bouret, P.. "Adaptive tests of homogeneity for a Poisson process." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 176-213. <http://eudml.org/doc/240914>.

@article{Fromont2011,
abstract = {We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in -norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.},
author = {Fromont, M., Laurent, B., Reynaud-Bouret, P.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Poisson process; adaptive hypotheses testing; uniform separation rate; minimax separation rate; model selection; thresholding rule},
language = {eng},
number = {1},
pages = {176-213},
publisher = {Gauthier-Villars},
title = {Adaptive tests of homogeneity for a Poisson process},
url = {http://eudml.org/doc/240914},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Fromont, M.
AU - Laurent, B.
AU - Reynaud-Bouret, P.
TI - Adaptive tests of homogeneity for a Poisson process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 1
SP - 176
EP - 213
AB - We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in -norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.
LA - eng
KW - Poisson process; adaptive hypotheses testing; uniform separation rate; minimax separation rate; model selection; thresholding rule
UR - http://eudml.org/doc/240914
ER -

References

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