Electronic Journal of Statistics

Detection of sparse additive functions

Ghislaine Gayraud and Yuri Ingster

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We study the problem of detection of high-dimensional signal functions in the Gaussian white noise model. We assume that, in addition to a smoothness assumption, the signal function has an additive sparse structure. The detection problem is expressed in terms of a nonparametric hypothesis testing problem and is solved using asymptotically minimax approach. We provide minimax test procedures that are adaptive in the sparsity parameter in the high sparsity case. We extend some known results related to the detection of sparse high-dimensional vectors to the functional case. In particular, our derivation of asymptotic detection rates is based on same detection boundaries as in the vector case.

Article information

Electron. J. Statist., Volume 6 (2012), 1409-1448.

First available in Project Euclid: 31 August 2012

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Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing 60G15: Gaussian processes 62G10: Hypothesis testing 62G20: Asymptotic properties 60C20

High-dimensional setting sparsity asymptotic minimax approach detection boundary Gaussian white noise model


Gayraud, Ghislaine; Ingster, Yuri. Detection of sparse additive functions. Electron. J. Statist. 6 (2012), 1409--1448. doi:10.1214/12-EJS715. https://projecteuclid.org/euclid.ejs/1346421599

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  • [1] Bickel, P.J., Ritov, Y. and Tsybakov, A.B. (2009). Simultaneous analysis of Lasso and Dantzig selector., Ann. Statist. 37 1705–1732.
  • [2] Cai, T., Jin, J. and Low, M. (2007). Estimation and confidence sets for sparse normal mixtures., Ann. Statist. 35 2421–2449.
  • [3] Donoho, D.L. (2006). Compressed Sensing., IEEE Transactions on Information Theory 52 1289–1306.
  • [4] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures., Ann. Statist. 32 962–994.
  • [5] Huang, J., Horowitz, J.L. and Wei, F. (2010). Variable selection in nonparametric additive models., Ann. Statist. 38 2282–2313.
  • [6] Ibragimov, I.A. and Khasminskii, R.Z. (1997). Some estimation problems on Infinite dimensional Gaussian white noise. In: Festschrift for Lucien Le Cam. Research papers in Probability and Statistics, 275–296. Springer-Verlag, New, York.
  • [7] Ingster, Yu.I. (1997). Some problems of hypothesis testing leading to infinitely divisible distributions., Math. Methods of Statist. 6 47–69.
  • [8] Ingster, Yu.I. (2001). Adaptive detection of a signal of growing dimension. I., Math. Methods Statist. 10 395–421.
  • [9] Ingster, Yu.I. (2002). Adaptive detection of a signal of growing dimension. II., Math. Methods Statist. 11 37–68.
  • [10] Ingster, Yu.I. and Lepski, O. (2003). On multichannel signal detection., Math. Methods Statist. 12 247–275.
  • [11] Ingster, Yu.I., Pouet, Ch. and Tsybakov, A.B. (2009). Classification of sparse high-dimensional vectors., Phi. Trans. R. Soc. A. 367 4427–4448.
  • [12] Ingster, Yu.I. and Suslina, I.A. (2002). On a detection of a signal of known shape in multichannel system., Zapiski Nauchn. Sem. POMI 294 88–112 (Transl. J. Math. Sci. (2005) 127 1723–1736).
  • [13] Ingster, Yu.I. and Suslina, I.A. (2003)., Nonparametric goodness-of-fit testing under gaussian models. Lectures Notes in Statistics, vol. 169. Springer-Verlag, New York.
  • [14] Ingster, Yu.I. and Suslina, I.A. (2005). On estimation and detection of smooth function of many variables., Math. Methods Statist. 14 299–331.
  • [15] Ingster, Yu.I. and Suslina, I.A. (2007). Estimation and detection of high-variable functions from Sloan-Woźniakowski space., Math. Methods Statist. 16 318–353.
  • [16] Ingster, Yu.I. and Suslina, I.A. (2007). On estimation and detection of a function from tensor product spaces. (In Russian)., Zapiski Nauchn. Sem. POMI 351 180–218 (Translation in: J. Math. Sci. (2008), 152 897–920).
  • [17] Ingster, Yu.I., Tsybakov, A.B. and Verzelen, N. (2010). Detection boundary in sparse regression., Electronic J. of Statistics 4 1476–1526.
  • [18] Lin, Y. (2000). Tensor product space ANOVA model., Ann. Statist. 28 734–755.
  • [19] Raskutti, G., Wainwright, M.J. and Yu, B. (2011). Minimax-optimal rates for sparse additive models over kernel classes via convex programming., http://arxiv.org/abs/1008.3654.
  • [20] Stone, Ch. (1985). Additive regression and other nonparametric models., Ann. Statist. 13 689–705.
  • [21] Tibshirani, R. (1996). Regression shrinkage and selection via Lasso., J. Roy. Statist. Soc. Ser. B. 58 267–288.