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 ${l}_{2}-$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.

Keywords: 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, Volume 10 (2006), pp. 46-75. doi : 10.1051/ps:2006002. http://www.numdam.org/articles/10.1051/ps:2006002/

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