Abstract
We consider minimax signal detection in the sequence model. Working with certain ellipsoids in the space of square-summable sequences of real numbers, with a ball of positive radius removed, we obtain upper and lower bounds for the minimax separation radius in the non-asymptotic framework, i.e., for a fixed value of the involved noise level. We use very weak assumptions on the noise (i.e., fourth moments are assumed to be uniformly bounded). In particular, we do not use any kind of Gaussian distribution or independence assumption on the noise. It is shown that the established minimax separation rates are not faster than the ones obtained in the classical sequence model (i.e., independent standard Gaussian noise) but, surprisingly, are of the same order as the minimax estimation rates in the classical setting. Under an additional condition on the noise, the classical minimax separation rates are also retrieved in benchmark well-posed and ill-posed inverse problems.
Similar content being viewed by others
References
Y. Baraud, “Non-Asymptotic Minimax Rates of Testing in Signal Detection”, Bernoulli 8 (5), 577–606 (2002).
N. Bissantz, G. Claeskens, H. Holzmann, and A. Munk, “Testing for lack of fit in inverse regression–with applications to biophotonic imaging”, J. Roy. Statist. Soc., Ser. B 71 (1), 25–48 (2009).
R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, in CMS Books inMathematics/Ouvrages de Mathé matiques de la SMC (Springer, 2016).
L. Cavalier, “Estimation in a Problemof Fractional Integration”, InverseProblems 20 (5), 1445–1454 (2004).
L. Cavalier, Inverse Problems in Statistics, in Lecture Notes Statist. Proc., Vol. 203: Inverse Problems and High-Dimensional Estimation (Springer, Heidelberg, 2011), pp. 3–96.
Yu. I. Ingster, T. Sapatinas, and I. A. Suslina, “Minimax Nonparametric Testing in a Problem Related to the Radon Transform”, Math. Methods Statist. 20 (4), 347–364 (2011).
Yu. I. Ingster, T. Sapatinas, and I. A. Suslina, “Minimax Signal Detection in Ill-Posed Inverse Problems”, Ann. Statist. 40, 1524–1549 (2012).
Yu. I. Ingster and I. A. Suslina, Nonparametric Goodness-of-Fit Testing Under Gaussian Models, in Lecture Notes in Statist. (Springer-Verlag, New York, 2003), Vol.169.
L. Isserlis, “On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables”, Biometrika 12 (1.2), 134–139 (1918).
I. M. Johnstone, “Wavelet Shrinkage for Correlated Data and Inverse Problems: Adaptivity Results”, Statistica Sinica 9 (1), 51–83 (1999).
B. Laurent, J.-M. Loubes, and C. Marteau, “Testing Inverse Problems: A Direct or an Indirect Problem?”, J. Statist. Planning and Inference 141 (5), 1849–1861 (2011).
B. Laurent, J.-M. Loubes, and C. Marteau, “Non-Asymptotic Minimax Rates of Testing in Signal Detection with Heterogeneous Variances”, Electronic J. Statist. 6, 91–122 (2012).
S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic Press, San Diego, 1999).
C. Marteau and P. Mathé, “General Regularization Schemes for Signal Detection in Inverse Problems”, Math. Methods Statist. 23 (3), 176–200 (2014).
C. Marteau and T. Sapatinas, “A Unified Treatment for Non-Asymptotic and Asymptotic Approaches to Minimax Signal Detection”, Statist. Surveys 9, 253–297 (2015).
A. B. Tsybakov, Introduction to Nonparametric Estimation, in Springer Series in Statist., Revised and extended from the 2004 French original (Springer, New York, 2009).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Marteau, C., Sapatinas, T. Minimax signal detection under weak noise assumptions. Math. Meth. Stat. 26, 282–298 (2017). https://doi.org/10.3103/S1066530717040032
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530717040032