Adaptive goodness-of-fit testing from indirect observations
Cristina Butucea, Catherine Matias, and Christophe Pouet
Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 2
(2009), 352-372.
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 H1 is expressed with respect to 
-norm (i.e. has the form 
). Our procedure is adaptive with respect to the unknown smoothness parameter τ 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.
Résumé
Dans un modèle de convolution, les observations sont des variables aléatoires réelles dont la distribution est la convoluée entre une densité inconnue f et une densité de bruit g supposée entièrement connue. Nous supposons que g est de régularité polynomiale. Nous proposons un test d’adéquation de l’hypothèse H0: f=f0 lorsque l’alternative H1 est exprimée à partir de la norme 
(i.e. de la forme 
). Cette procédure est adaptative par rapport au paramètre inconnu τ qui décrit la régularité de f. Nous obtenons différentes vitesses de test (ψn) en fonction du type de régularité de f0 (polynomiale ou bien exponentielle). L’adaptativité induit une perte sur la vitesse de test, perte qui est calculée grâce à un théorème de type Berry–Esseen non-uniforme pour des U-statistiques dégénérées. Dans le cas d’une régularité polynomiale pour f0, nous prouvons que cette perte est optimale. Soulignons que l’alternative peut éventuellement inclure des densités qui sont plus régulières que la densité à tester sous l’hypothèse nulle, ce qui est un point de vue nouveau pour les tests d’adaptation.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aihp/1241024673
Digital Object Identifier: doi:10.1214/08-AIHP166
Zentralblatt MATH identifier: 1168.62040
Mathematical Reviews number (MathSciNet): MR2521406
References
[1] C. Butucea. Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT 28(1) (2004) 9–26.
[2] C. Butucea. Goodness-of-fit testing and quadratic functional estimation from indirect observations. Ann. Statist. 35(5) (2007) 1907–1930.
[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.
[4] M. Fromont and B. Laurent. Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34(2) (2006) 680–720.
[5] G. Gayraud and C. Pouet. Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133(4) (2005) 531–558.
[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.
[7] P. Hall. Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal. 14(3) (1984) 1–16.
Mathematical Reviews (MathSciNet):
MR734096
[8] P. Hall and C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980.
Mathematical Reviews (MathSciNet):
MR624435
[9] H. Holzmann, N. Bissantz and A. Munk. Density testing in a contaminated sample. J. Multivariate Anal. 98(1) (2007) 57–75.
[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.
[11] Y. Ingster and I. Suslina. Nonparametric Goodness-Of-Fit Testing Under Gaussian Models. Springer, New York, 2003.
[12] V. Korolyuk and Y. Borovskikh. Theory of U-statistics. Kluwer Academic Publishers, Dordrecht, 1994.
[13] C. Pouet. On testing non-parametric hypotheses for analytic regression functions in Gaussian noise. Math. Methods Statist. 8(4) (1999) 536–549.
[14] V. Spokoiny. Adaptive hypothesis testing using wavelets. Ann. Statist. 24(6) (1996) 2477–2498.
