Electronic Journal of Statistics

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Béatrice Laurent, Jean-Michel Loubes, and Clément Marteau

Full-text: Open access

Abstract

The aim of this paper is to establish non-asymptotic minimax rates for goodness-of-fit hypotheses testing in an heteroscedastic setting. More precisely, we deal with sequences (Yj)jJ of independent Gaussian random variables, having mean (θj)jJ and variance (σj)jJ. The set J will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to l2 norm, without assumption on (σj)jJ and on several functions spaces. Our point of view is entirely non-asymptotic.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 91-122.

Dates
First available in Project Euclid: 3 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1328280899

Digital Object Identifier
doi:10.1214/12-EJS667

Mathematical Reviews number (MathSciNet)
MR2879673

Zentralblatt MATH identifier
1334.62085

Subjects
Primary: 62G05: Estimation 62K20: Response surface designs

Keywords
Goodness-of-fit tests heterogeneous variances inverse problems

Citation

Laurent, Béatrice; Loubes, Jean-Michel; Marteau, Clément. Non asymptotic minimax rates of testing in signal detection with heterogeneous variances. Electron. J. Statist. 6 (2012), 91--122. doi:10.1214/12-EJS667. https://projecteuclid.org/euclid.ejs/1328280899


Export citation

References

  • [1] Aldous, D. J. (1985), Exchangeability and related topics. Ecole d’été de probabilités de Saint-Flour XIII, Lect. Notes Math. 11117, 1–198.
  • [2] Baraud, Y. (2002), Non asymptotic minimax rates of testing in signal detection, Bernoulli, 8, 577–606.
  • [3] Baraud, Y., Huet, S. and Laurent, B. (2003), Adaptive tests of linear hypotheses by model selection, Ann. Statist., 31, no. 1, 225–251.
  • [4] Birgé, L. (2001), An alternative point of view on Lepski’s method, State of the art in probability and statistics (Leiden, 1999) (ed. Monogr., IMS Lecture Notes, 36, 113–133.
  • [5] Bissantz, N. and Claeskens, N. and Holzmann, H. and Munk, A. (2009), Testing for lack of fit in inverse regression, with applications to biophotonic imaging, J. R. Stat. Soc. Ser. B Stat. Methodol., 71, no. 1, 25–48.
  • [6] Butucea, C. (2007), Goodness-of-fit testing and quadratic functional estimation from indirect observations, Ann. Statist., 35, no. 5, 1907–1930.
  • [7] Cavalier, L. (2008), Nonparametric statistical inverse problems, Inverse Problems, 24(3).
  • [8] Donoho, D. (1995), Nonlinear solution of linear inverse problems by Wavelet-Vaguelette decomposition, Applied and computational harmonic analysis, 2, 101–126.
  • [9] Fromont, M., Laurent, B. and Reynaud-Bouret, P. (2011), Adaptive test of homogeneity for a Poisson process . Ann. Inst. H. Poincaré Probab. Statist., 47 no. 1, 176–213.
  • [10] Ermakov, M. S. (1990), Minimax detection of a signal in Gaussian white noise, Theory Probab. Appl., (35), 4, 667–679.
  • [11] Ermakov, M. S. (2006), Minimax detection of a signal in the heteroscedastic Gaussian white noise, J. Math. Sci., 137, No. 1, 4516–4524.
  • [12] Ingster, Yu. I. (1993), Asymptotically minimax testing for nonparametric alternatives I-II-III, Math. Methods Statist., 2, 85–114, 171–189, 249–268.
  • [13] Ingster, Yu. I., Sapatinas, T. and Suslina, I. A. (2010), Minimax signal detection in ill-posed inverse problems. arXiv:1001.1853.
  • [14] Ingster, Yu. I. and Suslina, I. A. (1998), Minimax detection of a signal for Besov bodies and balls, Problems Inform. Transmission, 34, 48–59.
  • [15] Ingster, Yu. I. and Suslina, I. A. (2002), Nonparametric goodness-of-fit testing under Gaussian models, Lecture Notes in Statistics, 169. Springer-Verlag, New York.
  • [16] Laurent, B. and Massart, P. (2000), Adaptive estimation of a quadratic functional by model selection, Ann. Statist., 28, no. 5, 1302–1338.
  • [17] Laurent, B., Loubes, J-M. and Marteau, C. (2011), Testing inverse problems: a direct or an indirect problem, J. Statist. Plann. Inference, 141, no. 5, 1849–1861.
  • [18] Lepski, O. V. and Spokoiny, V. G. (1999), Minimax nonparametric hypothesis testing: the case of inhomogeneous alternative, Bernoulli, 5, 333–358.
  • [19] Loubes, J-M. and Ludena, C. (2008), Adaptive complexity regularization for inverse problems, Electron. J. Stat., 2, 661–677.
  • [20] Loubes, J-M. and Ludena, C. (2010), Penalized estimators for non linear inverse problems, ESAIM Probab. Stat., 14, 173–191.
  • [21] O’Sullivan, F. (1986), A statistical perspective on ill-posed inverse problems, Statist. Sci., 1(4), 502–527.
  • [22] Spokoiny, V. G. (1996), Adaptive hypothesis testing using wavelets, Ann. Statist., 24, 2477–2498.