# Adaptive tests for periodic signal detection with applications to laser vibrometry

Magalie Fromont; Céline Lévy-leduc

ESAIM: Probability and Statistics (2006)

- Volume: 10, page 46-75
- ISSN: 1292-8100

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topFromont, Magalie, and Lévy-leduc, Céline. "Adaptive tests for periodic signal detection with applications to laser vibrometry." ESAIM: Probability and Statistics 10 (2006): 46-75. <http://eudml.org/doc/249739>.

@article{Fromont2006,

abstract = {
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
l2-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.
},

author = {Fromont, Magalie, Lévy-leduc, Céline},

journal = {ESAIM: Probability and Statistics},

keywords = { Periodic signal detection; adaptive test;
minimax separation rates; nonparametric regression.; periodic signal detection; minimax separation rates},

language = {eng},

month = {1},

pages = {46-75},

publisher = {EDP Sciences},

title = {Adaptive tests for periodic signal detection with applications to laser vibrometry},

url = {http://eudml.org/doc/249739},

volume = {10},

year = {2006},

}

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

DA - 2006/1//

PB - EDP Sciences

VL - 10

SP - 46

EP - 75

AB -
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
l2-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.

LA - eng

KW - Periodic signal detection; adaptive test;
minimax separation rates; nonparametric regression.; periodic signal detection; minimax separation rates

UR - http://eudml.org/doc/249739

ER -

## References

top- Y. Baraud, Non-asymptotic minimax rates of testing in signal detection. Bernoulli8 (2002) 577–606.
- Y. Baraud, S. Huet, and B. Laurent, Adaptive tests of linear hypotheses by model selection. Ann. Statist.31 (2003) 225–251.
- 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).
- P.J. Brockwell and R.A. Davis, Time series: theory and methods. Springer Series in Statistics. Springer-Verlag, New York, second edition (1991).
- R. Eubank and J. Hart, Testing goodness-of-fit in regression via order selection criteria. Ann. Stat.20 (1992) 1412–1425.
- J. Fan and Q. Yao, Nonlinear Time series. Springer series in Statistics. Springer-Verlag, New York, Nonparametric and parametric methods (2003).
- G. Gayraud and C. Pouet, Minimax testing composite null hypotheses in the discrete regression scheme. Math. Methods Stat.10 (2001) 375–394.
- 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.
- W. Härdle and A. Kneip, Testing a regression model when we have smooth alternatives in mind. Scand. J. Stat.26 (1999) 221–238.
- J. Horowitz and V. Spokoiny, An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica69 (2001) 599–631.
- Y. Ingster, Minimax nonparametric detection of signals in white Gaussian noise. Probl. Inf. Transm.18 (1982) 130–140.
- Y. Ingster, Asymptotically minimax testing for nonparametric alternatives I-II-III. Math. Methods Statist.2 (1993) 85–114, 171–189, 249–268.
- B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302–1338.
- 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.
- O. Lepski and V. Spokoiny, Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli5 (1999) 333–358.
- O. Lepski and A. Tsybakov, Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Relat. Fields117 (2000) 17–48.
- 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).
- 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).
- V. Spokoiny, Adaptive hypothesis testing using wavelets. Ann. Stat.24 (1996) 2477–2498.
- V. Spokoiny, Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Stat.7 (1998) 245–273.

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