Adaptive tests for periodic signal detection with applications to laser vibrometry

Abstract

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.Mathematics Subject Classification. 62G10, 62G08, 62G20.

Aggregated kernel based tests for signal detection in a regression model

“…2. Non asymptotic lower bounds for the rates of testing in signal detection over Sobolev balls are given in Fromont and Lévy-Leduc (2006). These bounds coincide with the bound given in (25).…”

Adaptive minimax testing in the discrete regression scheme

“…Other papers must be mentioned here in the context of adaptive hypothesis testing about the regression function : in Baraud, Huet and Laurent (2003), no assumption on f is required since the distance they used to separate the null hypothesis and the alternative is a discrete one; it avoids to quantify the approximation of the L 2 -norm, when it is used as the distance as in our study, by a discrete sum of squared terms. The same occurs in Horowitz and Spokoiny (2001) and in Fromont and Lévy-Leduc (2003). Horowitz and Spokoiny (2001) give results for a composite null hypothesis and for Hölder spaces.…”

“…But we are more general than Horowitz and Spokoiny (2001) in the choice of the class under the null hypothesis since they take a parametric family of given functions whereas in our study only a control of the entropy of the class which could even be nonparametric is required. Fromont and Lévy-Leduc (2003) consider the problem of periodic signal detection in a Gaussian fixed design regression framework, assuming that the signal belongs to some periodic Sobolev balls. Fan, Zhang and Zhang (2001) give adaptive results when the alternatives lie in a range of Sobolev ball; their test statistic is based on generalized likelihood ratio and they obtain the asymptotic distribution of their test statistic under the null hypothesis, which is free of any nuisance parameter (this result is referred as Wilks phenomenon).…”

Minimax hypothesis testing for curve registration

“…In the regression and Gaussian sequence models, Baraud [1] derived nonasymptotic minimax separation rates when the functions in the alternative lie in l p -bodies (0 < p ≤ 2) and the separation from 0 is measured by the l 2norm. Baraud, Huet, and Laurent [2,3] proposed procedures for testing linear or convex hypotheses in the regression model, and Fromont and Lévy-Leduc [18] inspected the improvement implied by a further hypothesis on the periodicity of the signal in the periodic Sobolev balls.…”