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Adaptive minimax testing in the discrete regression scheme
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  • Published: 06 June 2005

Adaptive minimax testing in the discrete regression scheme

  • Ghislaine Gayraud1,2 &
  • Christophe Pouet3 

Probability Theory and Related Fields volume 133, pages 531–558 (2005)Cite this article

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  • 16 Citations

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Abstract

We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0,1]. We wish to test the null hypothesis that the regression function belongs to a given functional class (parametric or even nonparametric) against a composite nonparametric alternative. The functions under the alternative are separated in the L2-norm from any function in the null hypothesis. We assume that the regression function belongs to a wide range of Hölder classes but as the smoothness parameter of the regression function is unknown, an adaptive approach is considered. It leads to an optimal and unavoidable loss of order in the minimax rate of testing compared with the non-adaptive setting. We propose a smoothness-free test that achieves the optimal rate, and finally we prove the lower bound showing that no test can be consistent if in the distance between the functions under the null hypothesis and those in the alternative, the loss is of order smaller than the optimal loss.

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Author information

Authors and Affiliations

  1. CREST, Timbre J340, 3 av. P. Larousse, 92241, Malakoff Cedex

    Ghislaine Gayraud

  2. Laboratoire de Mathématiques Raphael Salem, Université de Rouen, France

    Ghislaine Gayraud

  3. Centre de Mathématiques et Informatique, Université de Provence, 39, rue Joliot Curie, 13453, Marseille Cedex 13, France

    Christophe Pouet

Authors
  1. Ghislaine Gayraud
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  2. Christophe Pouet
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Correspondence to Christophe Pouet.

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Cite this article

Gayraud, G., Pouet, C. Adaptive minimax testing in the discrete regression scheme. Probab. Theory Relat. Fields 133, 531–558 (2005). https://doi.org/10.1007/s00440-005-0445-4

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  • Received: 19 August 2003

  • Revised: 24 February 2005

  • Published: 06 June 2005

  • Issue Date: December 2005

  • DOI: https://doi.org/10.1007/s00440-005-0445-4

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Mathematics Subject Classification (2000)

  • 62G10
  • 62G08; Secondary 62C20

Keywords

  • Adaptive testing
  • Composite null hypothesis
  • Nonparametric hypotheses testing
  • Minimax rate of testing
  • Regression model
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