Electronic Journal of Statistics

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Gilles Blanchard, Alexandra Carpentier, and Maurilio Gutzeit

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We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset $\mathcal{C}$ of $\mathbb{R}^{d}$. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension $d$ and variance $\frac{1}{n}$ giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for $\mathcal{C}$.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 3713-3735.

Received: February 2017
First available in Project Euclid: 7 November 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Minimax hypothesis testing Gaussian sequence model nonasymptotic minimax separation rate

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Blanchard, Gilles; Carpentier, Alexandra; Gutzeit, Maurilio. Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$. Electron. J. Statist. 12 (2018), no. 2, 3713--3735. doi:10.1214/18-EJS1472. https://projecteuclid.org/euclid.ejs/1541559861

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  • [1] Arias-Castro, E., and Casal, A. R. On estimating the perimeter using the alpha-shape., Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53, 3 (2017), 1051–1068.
  • [2] Baraud, Y. Non-asymptotic minimax rates of testing in signal detection., Bernoulli 8, 5 (2002), 577–606.
  • [3] Baraud, Y., Huet, S., and Laurent, B. Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function., The Annals of Statistics (2005), 214–257.
  • [4] Birgé, L. An alternative point of view on Lepski’s method., Lecture Notes-Monograph Series (2001), 113–133.
  • [5] Bull, A., and Nickl, R. Adaptive confidence sets in $l_2$., Probability Theory and Related Fields 156, 3-4 (2013), 889–919.
  • [6] Burnashev, M. On the minimax detection of an imperfectly known signal in a white noise background., Theory Probab. Appl. 24 (1979), 107–119.
  • [7] Cai, T. T., and Low, M. Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional., The Annals of Statistics 39, 2 (2011), 1012–1041.
  • [8] Carpentier, A. Testing the regularity of a smooth signal., Bernoulli 21, 1 (2015), 465–488.
  • [9] Casey, J. E xploring Curvature., Vieweg Wiesbaden, 1996.
  • [10] Chernoff, H. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations., The Annals of Mathematical Statistics 23 (1952), 493–507.
  • [11] Comminges, L., and Dalalyan, A. Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression., Electronic Journal of Statistics 7 (2013), 146–190.
  • [12] Gayraud, G., and Pouet, C. Adaptive minimax testing in the discrete regression scheme., Probability Theory and Related Fields 133, 4 (2005), 531–558.
  • [13] Ingster, Y. On testing a hypothesis which is close to a simple hypothesis., Theory Prob. Appl. 45 (2000), 310–323.
  • [14] Ingster, Y., and Suslina, I. Minimax detection of a signal for Besov bodies and balls., Problems of Information Transmission 34, 1 (1998), 48–59.
  • [15] Ingster, Y., and Suslina, I., Nonparametric goodness-of-fit testing under Gaussian models. Springer-Verlag New York, Inc., 2003.
  • [16] Juditsky, A., and Nemirovski, A. On nonparametric tests of positivity/monotonicity/convexity., The Annals of Statistics (2002), 498–527.
  • [17] Lepski, O., Nemirovski, A., and Spokoiny, V. On estimation of the $l_r$ norm of a regression function., Probability Theory and Related Fields 113, 2 (1999), 221–253.