Adaptive goodness-of-fit testing from indirect observations

Cristina Butucea; Catherine Matias; Christophe Pouet

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 2, page 352-372
  • ISSN: 0246-0203

Abstract

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In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0: f=f0, where the alternative H1is expressed with respect to 𝕃 2 -norm (i.e. has the form ψ n - 2 ∥ f - f 0 ∥ 2 2 ≥ 𝒞 ). Our procedure is adaptive with respect to the unknown smoothness parameterτ of f. Different testing rates (ψn) are obtained according to whether f0 is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry–Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f0, we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests.

How to cite

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Butucea, Cristina, Matias, Catherine, and Pouet, Christophe. "Adaptive goodness-of-fit testing from indirect observations." Annales de l'I.H.P. Probabilités et statistiques 45.2 (2009): 352-372. <http://eudml.org/doc/78026>.

@article{Butucea2009,
abstract = {In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0: f=f0, where the alternative H1is expressed with respect to $\mathbb \{L\}_\{2\}$-norm (i.e. has the form $\psi _\{n\}^\{-2\}\Vert f-f_\{0\}\Vert _\{2\}^\{2\}\ge \mathcal \{C\}$). Our procedure is adaptive with respect to the unknown smoothness parameterτ of f. Different testing rates (ψn) are obtained according to whether f0 is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry–Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f0, we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests.},
author = {Butucea, Cristina, Matias, Catherine, Pouet, Christophe},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {adaptive nonparametric tests; convolution model; goodness-of-fit tests; infinitely differentiable functions; partially known noise; quadratic functional estimation; Sobolev classes; stable laws},
language = {eng},
number = {2},
pages = {352-372},
publisher = {Gauthier-Villars},
title = {Adaptive goodness-of-fit testing from indirect observations},
url = {http://eudml.org/doc/78026},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Butucea, Cristina
AU - Matias, Catherine
AU - Pouet, Christophe
TI - Adaptive goodness-of-fit testing from indirect observations
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 2
SP - 352
EP - 372
AB - In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0: f=f0, where the alternative H1is expressed with respect to $\mathbb {L}_{2}$-norm (i.e. has the form $\psi _{n}^{-2}\Vert f-f_{0}\Vert _{2}^{2}\ge \mathcal {C}$). Our procedure is adaptive with respect to the unknown smoothness parameterτ of f. Different testing rates (ψn) are obtained according to whether f0 is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry–Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f0, we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests.
LA - eng
KW - adaptive nonparametric tests; convolution model; goodness-of-fit tests; infinitely differentiable functions; partially known noise; quadratic functional estimation; Sobolev classes; stable laws
UR - http://eudml.org/doc/78026
ER -

References

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  10. [10] C. Houdré and P. Reynaud-Bouret. Exponential inequalities, with constants, for U-statistics of order two. In Stochastic Inequalities and Applications 55–69. Progr. Probab. 56. Birkhäuser, Basel, 2003. Zbl1036.60015MR2073426
  11. [11] Y. Ingster and I. Suslina. Nonparametric Goodness-Of-Fit Testing Under Gaussian Models. Springer, New York, 2003. Zbl1013.62049MR1991446
  12. [12] V. Korolyuk and Y. Borovskikh. Theory of U-statistics. Kluwer Academic Publishers, Dordrecht, 1994. Zbl0785.60015MR1472486
  13. [13] C. Pouet. On testing non-parametric hypotheses for analytic regression functions in Gaussian noise. Math. Methods Statist. 8(4) (1999) 536–549. Zbl1103.62344MR1755899
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