Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.
Mots clés : periodic signal detection, adaptive test, minimax separation rates, nonparametric regression
@article{PS_2006__10__46_0,
author = {Fromont, Magalie and L\'evy-Leduc, C\'eline},
title = {Adaptive tests for periodic signal detection with applications to laser vibrometry},
journal = {ESAIM: Probability and Statistics},
pages = {46--75},
publisher = {EDP-Sciences},
volume = {10},
year = {2006},
doi = {10.1051/ps:2006002},
mrnumber = {2197102},
zbl = {1141.62302},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2006002/}
}
TY - JOUR AU - Fromont, Magalie AU - Lévy-Leduc, Céline TI - Adaptive tests for periodic signal detection with applications to laser vibrometry JO - ESAIM: Probability and Statistics PY - 2006 SP - 46 EP - 75 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006002/ DO - 10.1051/ps:2006002 LA - en ID - PS_2006__10__46_0 ER -
%0 Journal Article %A Fromont, Magalie %A Lévy-Leduc, Céline %T Adaptive tests for periodic signal detection with applications to laser vibrometry %J ESAIM: Probability and Statistics %D 2006 %P 46-75 %V 10 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2006002/ %R 10.1051/ps:2006002 %G en %F PS_2006__10__46_0
Fromont, Magalie; Lévy-Leduc, Céline. Adaptive tests for periodic signal detection with applications to laser vibrometry. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 46-75. doi : 10.1051/ps:2006002. http://www.numdam.org/articles/10.1051/ps:2006002/
[1] , Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 (2002) 577-606. | Zbl
[2] ,, and, Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (2003) 225-251. | Zbl
[3] , An alternative point of view on Lepski's method, in State of the Art in Probability and Statistics (Leiden, 1999), 113-133, IMS Lecture Notes Monogr. Ser. 36 (2000).
[4] and, Time series: theory and methods. Springer Series in Statistics. Springer-Verlag, New York, second edition (1991). | MR | Zbl
[5] and, Testing goodness-of-fit in regression via order selection criteria. Ann. Stat. 20 (1992) 1412-1425. | Zbl
[6] and, Nonlinear Time series. Springer series in Statistics. Springer-Verlag, New York, Nonparametric and parametric methods (2003). | MR | Zbl
[7] and, Minimax testing composite null hypotheses in the discrete regression scheme. Math. Methods Stat. 10 (2001) 375-394. | Zbl
[8] and, A new method for the detection of a periodic signal of unknown shape and period. The Astrophysical J. 398 (1992) 146-168.
[9] and, Testing a regression model when we have smooth alternatives in mind. Scand. J. Stat. 26 (1999) 221-238. | Zbl
[10] and, An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 (2001) 599-631. | Zbl
[11] , Minimax nonparametric detection of signals in white Gaussian noise. Probl. Inf. Transm. 18 (1982) 130-140. | Zbl
[12] , Asymptotically minimax testing for nonparametric alternatives I-II-III. Math. Methods Statist. 2 (1993) 85-114, 171-189, 249-268. | Zbl
[13] and, Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 (2000) 1302-1338. | Zbl
[14] and, Semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals. IEEE Trans. Signal Proces. 53 (2005) 2306-2314.
[15] and, Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli 5 (1999) 333-358. | Zbl
[16] and, Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Relat. Fields 117 (2000) 17-48. | Zbl
[17] , Vibration modes and laser vibrometry performance in noise, in Proceedings of the Physics in Signal and Image Processing conference (PSIP'01), 23-24 janvier 2001, Marseille, France (2001).
[18] and, The estimation and tracking of frequency. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2001). | MR | Zbl
[19] , Adaptive hypothesis testing using wavelets. Ann. Stat. 24 (1996) 2477-2498. | Zbl
[20] , Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Stat. 7 (1998) 245-273. | Zbl
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