# Adaptive tests of qualitative hypotheses

Yannick Baraud; Sylvie Huet; Béatrice Laurent

ESAIM: Probability and Statistics (2010)

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

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topBaraud, 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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- Y. Baraud, S. Huet and B. Laurent, Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (2003). Zbl1018.62037
- Y. Baraud, S. Huet and B. Laurent, Tests for convex hypotheses, Technical Report 2001-66. University of Paris XI, France (2001). Zbl1014.62052
- H.D. Brunk, On the estimation of parameters restricted by inequalities. Ann. Math. Statist.29 (1958) 437-454. Zbl0087.34302
- L. Dümbgen and V.G. Spokoïny, Multiscale testing of qualitative hypotheses. Ann. Statist.29 (2001) 124-152. Zbl1029.62070
- S. Ghosal, A. Sen and A. van der Vaart, Testing monotonicity of regression. Ann. Statist.28 (2000) 1054-1082.
- I. Gijbels, P. Hall, M.C. Jones and I. Koch, Tests for monotonicity of a regression mean with guaranteed level. Biometrika87 (2000) 663-673. Zbl0956.62039
- P. Hall and N. Heckman, Testing for monotonicity of a regression mean by calibrating for linear functions. Ann. Statist.28 (2000) 20-39. Zbl1106.62324
- I.A. Ibragimov and R.Z. Has'minskii, Statistical estimation. Asymptotic theory. Springer-Verlag, New York-Berlin, Appl. Math. 16 (1981).
- A. Juditsky and A. Nemirovski, On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist.30 (2002) 498-527. Zbl1012.62048
- B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302-1338. Zbl1105.62328

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