# 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

## Access Full Article

top## Abstract

top## How to cite

topButucea, 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

top- [1] C. Butucea. Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT 28(1) (2004) 9–26. Zbl1274.62229MR2076033
- [2] C. Butucea. Goodness-of-fit testing and quadratic functional estimation from indirect observations. Ann. Statist. 35(5) (2007) 1907–1930. Zbl1126.62028MR2363957
- [3] C. Butucea, C. Matias and C. Pouet. Adapativity in convolution models with partially known noise distribution. Technical report, 2007. Available at http://arxiv.org/abs/0804.1056. Zbl1320.62066MR2447344
- [4] M. Fromont and B. Laurent. Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34(2) (2006) 680–720. Zbl1096.62040MR2281881
- [5] G. Gayraud and C. Pouet. Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133(4) (2005) 531–558. Zbl1075.62029MR2197113
- [6] E. Giné, R. Latała and J. Zinn. Exponential and moment inequalities for U-statistics. In High Dimensional Probability, II (Seattle, WA, 1999) 13–38. Progr. Probab. 47. Birkhäuser Boston, Boston, MA, 2000. Zbl0969.60024MR1857312
- [7] P. Hall. Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal. 14(3) (1984) 1–16. Zbl0528.62028MR734096
- [8] P. Hall and C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980. Zbl0462.60045MR624435
- [9] H. Holzmann, N. Bissantz and A. Munk. Density testing in a contaminated sample. J. Multivariate Anal. 98(1) (2007) 57–75. Zbl1102.62045MR2292917
- [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] Y. Ingster and I. Suslina. Nonparametric Goodness-Of-Fit Testing Under Gaussian Models. Springer, New York, 2003. Zbl1013.62049MR1991446
- [12] V. Korolyuk and Y. Borovskikh. Theory of U-statistics. Kluwer Academic Publishers, Dordrecht, 1994. Zbl0785.60015MR1472486
- [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
- [14] V. Spokoiny. Adaptive hypothesis testing using wavelets. Ann. Statist. 24(6) (1996) 2477–2498. Zbl0898.62056MR1425962

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.