Adaptive tests for periodic signal detection with applications to laser vibrometry
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 46-75.

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.

DOI : https://doi.org/10.1051/ps:2006002
Classification : 62G10,  62G08,  62G20
Mots clés : periodic signal detection, adaptive test, minimax separation rates, nonparametric regression
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author = {Fromont, Magalie and L\'evy-Leduc, C\'eline},
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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] Y. Baraud, Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 (2002) 577-606. | Zbl 1007.62042

[2] Y. Baraud, S. Huet, and B. Laurent, Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (2003) 225-251. | Zbl 1018.62037

[3] L. Birgé, 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] P.J. Brockwell and R.A. Davis, Time series: theory and methods. Springer Series in Statistics. Springer-Verlag, New York, second edition (1991). | MR 1093459 | Zbl 0709.62080

[5] R. Eubank and J. Hart, Testing goodness-of-fit in regression via order selection criteria. Ann. Stat. 20 (1992) 1412-1425. | Zbl 0776.62045

[6] J. Fan and Q. Yao, Nonlinear Time series. Springer series in Statistics. Springer-Verlag, New York, Nonparametric and parametric methods (2003). | MR 1964455 | Zbl 1014.62103

[7] G. Gayraud and C. Pouet, Minimax testing composite null hypotheses in the discrete regression scheme. Math. Methods Stat. 10 (2001) 375-394. | Zbl 1005.62048

[8] P. Gregory and T. Loredo, A new method for the detection of a periodic signal of unknown shape and period. The Astrophysical J. 398 (1992) 146-168.

[9] W. Härdle and A. Kneip, Testing a regression model when we have smooth alternatives in mind. Scand. J. Stat. 26 (1999) 221-238. | Zbl 0934.62043

[10] J. Horowitz and V. Spokoiny, An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 (2001) 599-631. | Zbl 1017.62012

[11] Y. Ingster, Minimax nonparametric detection of signals in white Gaussian noise. Probl. Inf. Transm. 18 (1982) 130-140. | Zbl 0499.94002

[12] Y. Ingster, Asymptotically minimax testing for nonparametric alternatives I-II-III. Math. Methods Statist. 2 (1993) 85-114, 171-189, 249-268. | Zbl 0798.62059

[13] B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 (2000) 1302-1338. | Zbl 1105.62328

[14] M. Lavielle and C. Lévy-Leduc, 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] O. Lepski and V. Spokoiny, Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli 5 (1999) 333-358. | Zbl 0946.62050

[16] O. Lepski and A. Tsybakov, Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Relat. Fields 117 (2000) 17-48. | Zbl 0971.62022

[17] M. Prenat, 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] B.G. Quinn and E.J. Hannan, The estimation and tracking of frequency. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2001). | MR 1813156 | Zbl 0969.62060

[19] V. Spokoiny, Adaptive hypothesis testing using wavelets. Ann. Stat. 24 (1996) 2477-2498. | Zbl 0898.62056

[20] V. Spokoiny, Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Stat. 7 (1998) 245-273. | Zbl 1103.62345

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