Adaptive tests of qualitative hypotheses

Yannick Baraud; Sylvie Huet; Béatrice Laurent

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 147-159
  • ISSN: 1292-8100

Abstract

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We propose a test of a qualitative hypothesis on the mean of a n-Gaussian vector. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are non-asymptotic. For testing positivity or monotonicity, we establish separation rates with respect to the Euclidean distance, over subsets of ℝ n which are related to Hölderian balls in functional spaces. We provide a simulation study in order to evaluate the procedure when the purpose is to test monotonicity in a functional regression model and to check the robustness of the procedure to non-Gaussian errors.

How to cite

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Baraud, Yannick, Huet, Sylvie, and Laurent, Béatrice. "Adaptive tests of qualitative hypotheses." ESAIM: Probability and Statistics 7 (2010): 147-159. <http://eudml.org/doc/104300>.

@article{Baraud2010,
abstract = { We propose a test of a qualitative hypothesis on the mean of a n-Gaussian vector. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are non-asymptotic. For testing positivity or monotonicity, we establish separation rates with respect to the Euclidean distance, over subsets of $\mathbb\{R\}^\{n\}$ which are related to Hölderian balls in functional spaces. We provide a simulation study in order to evaluate the procedure when the purpose is to test monotonicity in a functional regression model and to check the robustness of the procedure to non-Gaussian errors. },
author = {Baraud, Yannick, Huet, Sylvie, Laurent, Béatrice},
journal = {ESAIM: Probability and Statistics},
keywords = {Adaptive test; test of monotonicity; test of positivity; qualitative hypothesis testing; nonparametric alternative; nonparametric regression.; adaptive test; qualitative hypothesis testing; nonparametric alternatives; nonparametric regression; simulation},
language = {eng},
month = {3},
pages = {147-159},
publisher = {EDP Sciences},
title = {Adaptive tests of qualitative hypotheses},
url = {http://eudml.org/doc/104300},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Baraud, Yannick
AU - Huet, Sylvie
AU - Laurent, Béatrice
TI - Adaptive tests of qualitative hypotheses
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 147
EP - 159
AB - We propose a test of a qualitative hypothesis on the mean of a n-Gaussian vector. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are non-asymptotic. For testing positivity or monotonicity, we establish separation rates with respect to the Euclidean distance, over subsets of $\mathbb{R}^{n}$ which are related to Hölderian balls in functional spaces. We provide a simulation study in order to evaluate the procedure when the purpose is to test monotonicity in a functional regression model and to check the robustness of the procedure to non-Gaussian errors.
LA - eng
KW - Adaptive test; test of monotonicity; test of positivity; qualitative hypothesis testing; nonparametric alternative; nonparametric regression.; adaptive test; qualitative hypothesis testing; nonparametric alternatives; nonparametric regression; simulation
UR - http://eudml.org/doc/104300
ER -

References

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  1. Y. Baraud, Model selection for regression on a fixed design. Probab. Theory Related Fields117 (2000) 467-493.  
  2. Y. Baraud, S. Huet and B. Laurent, Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (2003).  
  3. Y. Baraud, S. Huet and B. Laurent, Tests for convex hypotheses, Technical Report 2001-66. University of Paris XI, France (2001).  
  4. H.D. Brunk, On the estimation of parameters restricted by inequalities. Ann. Math. Statist.29 (1958) 437-454.  
  5. L. Dümbgen and V.G. Spokoïny, Multiscale testing of qualitative hypotheses. Ann. Statist.29 (2001) 124-152.  
  6. S. Ghosal, A. Sen and A. van der Vaart, Testing monotonicity of regression. Ann. Statist.28 (2000) 1054-1082.  
  7. I. Gijbels, P. Hall, M.C. Jones and I. Koch, Tests for monotonicity of a regression mean with guaranteed level. Biometrika87 (2000) 663-673.  
  8. P. Hall and N. Heckman, Testing for monotonicity of a regression mean by calibrating for linear functions. Ann. Statist.28 (2000) 20-39.  
  9. I.A. Ibragimov and R.Z. Has'minskii, Statistical estimation. Asymptotic theory. Springer-Verlag, New York-Berlin, Appl. Math. 16 (1981).  
  10. A. Juditsky and A. Nemirovski, On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist.30 (2002) 498-527.  
  11. B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302-1338.  

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