• Bernoulli
  • Volume 21, Number 1 (2015), 465-488.

Testing the regularity of a smooth signal

Alexandra Carpentier

Full-text: Open access


We develop a test to determine whether a function lying in a fixed $L_{2}$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in fact of a given smoothness $s\geq t$ or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness $t$, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness $t$. The functions that we remove are functions of smoothness strictly smaller than $s$, but that are very close to $s$-smooth functions. A lower bound on the size of this region has been proved to be of order $n^{-t/(2t+1/2)}$, and in this paper, we provide a test that is consistent after the removal of a region of such a size. Even though the null hypothesis is composite, the size of the region we remove does not depend on the complexity of the null hypothesis.

Article information

Bernoulli, Volume 21, Number 1 (2015), 465-488.

First available in Project Euclid: 17 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

functional analysis minimax bounds non-parametric composite testing problem


Carpentier, Alexandra. Testing the regularity of a smooth signal. Bernoulli 21 (2015), no. 1, 465--488. doi:10.3150/13-BEJ575.

Export citation


  • [1] Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528–551.
  • [2] Baraud, Y., Huet, S. and Laurent, B. (2005). Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function. Ann. Statist. 33 214–257.
  • [3] Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301–413.
  • [4] Belitser, E. and Enikeeva, F. (2008). Empirical Bayesian test of the smoothness. Math. Methods Statist. 17 1–18.
  • [5] Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction. Berlin: Springer.
  • [6] Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M. and Mikhaĭlovich, S. (1978). Integral Representations of Functions and Imbedding Theorems, Vol. 1. New York: Halsted Press.
  • [7] Birgé, L. (2001). An alternative point of view on Lepski’s method. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes – Monograph Series 36 113–133. Beachwood, OH: IMS.
  • [8] Blanchard, G., Delattre, S. and Roquain, E. (2014). Testing over a continuum of null hypotheses with False Discovery Rate control. Bernoulli 20 304–333.
  • [9] Bull, A.D. and Nickl, R. (2013). Adaptive confidence sets in $L^{2}$. Probab. Theory Related Fields 156 889–919.
  • [10] Cai, T.T. and Low, M.G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805–1840.
  • [11] Cai, T.T. and Low, M.G. (2006). Adaptive confidence balls. Ann. Statist. 34 202–228.
  • [12] Cohen, A., Daubechies, I. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 54–81.
  • [13] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • [14] Dümbgen, L. and Spokoiny, V.G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124–152.
  • [15] Dziedziul, K. and Ćmiel, B. (2014). Density smoothness estimation problem using a wavelet approach. ESAIM Probab. Statist. 18 130–144.
  • [16] Fromont, M. and Laurent, B. (2006). Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 680–720.
  • [17] Gayraud, G. and Pouet, C. (2005). Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133 531–558.
  • [18] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129. New York: Springer.
  • [19] Hoffmann, M. (1999). On nonparametric estimation in nonlinear $\operatorname{AR}(1)$-models. Statist. Probab. Lett. 44 29–45.
  • [20] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression. Ann. Statist. 30 325–396.
  • [21] Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands. Ann. Statist. 39 2383–2409.
  • [22] Horowitz, J.L. and Spokoiny, V.G. (2001). An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 599–631.
  • [23] Ingster, Yu.I. (1986). Minimax testing of nonparametric hypotheses on a distribution density in the $l_{p}$ metrics. Theory Probab. Appl. 31 333–337.
  • [24] Ingster, Yu.I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85–114.
  • [25] Ingster, Yu.I. and Suslina, I.A. (2003). Nonparametric Goodness-of-fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. New York: Springer.
  • [26] Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 12 410–428.
  • [27] Juditsky, A. and Nemirovski, A. (2002). On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist. 30 498–527.
  • [28] Lepski, O.V. and Spokoiny, V.G. (1999). Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli 5 333–358.
  • [29] Lepskiĭ, O.V. (1992). On problems of adaptive estimation in white Gaussian noise. In Topics in Nonparametric Estimation. Adv. Soviet Math. 12 87–106. Providence, RI: Amer. Math. Soc.
  • [30] Low, M.G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547–2554.
  • [31] Y. Meyer (1992). Wavelets and applications. In Proceedings of the Second International Conference Held in Marseille, May 1989. RMA: Research Notes in Applied Mathematics 20. Paris: Masson.
  • [32] Nickl, R. and van de Geer, S. (2013). Confidence sets in sparse regression. Ann. Statist. 41 2852–2876.
  • [33] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • [34] Pouet, C. (2002). Test asymptotiquement minimax pour une hypothèse nulle composite dans le modèle de densité. C. R. Math. Acad. Sci. Paris 334 913–916.
  • [35] Reiß, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Statist. 36 1957–1982.
  • [36] Robins, J. and van der Vaart, A. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • [37] Spokoiny, V.G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
  • [38] Tsybakov, A.B. (2004). Introduction à L’estimation Non-paramétrique. Mathématiques & Applications (Berlin) [Mathematics & Applications] 41. Berlin: Springer.