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Honest and adaptive confidence sets in \(L_p\). (English) Zbl 1280.62055
Summary: We consider the problem of constructing honest and adaptive confidence sets in \(L_p\)-loss (with \(p\geq 1\) and \(p < \infty\)) over sets of Sobolev-type classes, in the setting of nonparametric Gaussian regression. The objective is to adapt the diameter of the confidence sets with respect to the smoothness degree of the underlying function, while ensuring that the true function lies in the confidence interval with high probability. When \(p \geq 2\), we identify two main regimes, (i) one where adaptation is possible without any restrictions on the model, and (ii) one where critical regions have to be removed. We also prove by a matching lower bound that the size of the regions that we remove can not be chosen significantly smaller. These regimes are shown to depend in a qualitative way on the index \(p\), and a continuous transition from \(p = 2\) to \(p = \infty\) is exhibited.

MSC:
62G15 Nonparametric tolerance and confidence regions
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
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References:
[1] Baraud, Y. Confidence balls in gaussian regression. Annals of Statistics , pages 528-551, 2004. · Zbl 1093.62051
[2] Barron, A., Birgé, L., and Massart, P. Risk bounds for model selection via penalization. Probability Theory and Related Fields , 113(3):301-413, 1999. · Zbl 0946.62036
[3] Bergh, J. and Löfström, J. Interpolation Spaces: An Introduction , volume 223. Springer-verlag Berlin, 1976. · Zbl 0344.46071
[4] Besov, O.V., Il’in, V.P., and Nikol’skiĭ. Integral Representations of Functions and Imbedding Theorems .
[5] L. Birgé and Massart, P. Gaussian model selection. Journal of the European Mathematical Society , 3(3):203-268, 2001. · Zbl 1037.62001
[6] Bull, A.D. and Nickl, R. Adaptive confidence sets in \(L_{2}\). Probability Theory and Related Fields , 156(3):889-919, 2013. · Zbl 1273.62105
[7] Cai, T.T. and Low, M.G. Adaptive confidence balls. The Annals of Statistics , 34(1):202-228, 2006. · Zbl 1091.62037
[8] Carpentier, A. Testing the regularity of a smooth signal. To appear in Bernoulli , 2013. · Zbl 1320.94021
[9] Cohen, A., Daubechies, I., and Vial, P. Wavelets on the interval and fast wavelet transforms. Applied Computational Harmonic Analysis , 1(1):54-81, 1993. · Zbl 0795.42018
[10] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. Wavelet shrinkage: asymptopia? Journal of the Royal Statistical Society. Series B (Methodological) , pages 301-369, 1995. · Zbl 0827.62035
[11] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. Density estimation by wavelet thresholding. The Annals of Statistics , pages 508-539, 1996. · Zbl 0860.62032
[12] Efromovich, S. Adaptive estimation of and oracle inequalities for probability densities and characteristic functions. The Annals of Statistics , 36(3):1127-1155, 2008. · Zbl 1360.62118
[13] Giné, E. and Nickl, R. Uniform limit theorems for wavelet density estimators. The Annals of Probability , 37(4):1605-1646, 2009b. · Zbl 1255.62103
[14] Giné, E. and Nickl, R. Confidence bands in density estimation. The Annals of Statistics , 38(2):1122-1170, 2010b. · Zbl 1183.62062
[15] Giné, E. and Nickl, R. Rates of contraction for posterior distributions in \(l_{r}\)-metrics, \(\{1\leq r\leq+\infty\}\). The Annals of Statistics , 39(6):2883-2911, 2011. · Zbl 1246.62095
[16] Härdle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A. Wavelets, Approximation, and Statistical Applications . Springer, New York, 1998. · Zbl 0899.62002
[17] Hoffman, M. and Lepski, O. Random rates in anisotropic regression (with a discussion and a rejoinder by the authors). The Annals of Statistics , 30(2):325-396, 2002. · Zbl 1012.62042
[18] Hoffmann, M. and Nickl, R. On adaptive inference and confidence bands. The Annals of Statistics , 39(5):2383-2409, 2011. · Zbl 1232.62072
[19] Ingster, Y. and Suslina, I.A. Nonparametric Goodness-of-fit Testing Under Gaussian Models , volume 169. Springer, 2002. · Zbl 1013.62049
[20] Ingster, Y.I. Minimax testing of nonparametric hypotheses on a distribution density in the \(l_{p}\) metrics. Theory of Probability & Its Applications , 31(2):333-337, 1987. · Zbl 0629.62049
[21] Ingster, Y.I. Asymptotically minimax hypothesis testing for nonparametric alternatives. i, ii, iii. Math. Methods Statist. , 2(2):85-114, 1993. · Zbl 0798.62057
[22] Juditsky, A. and S. Lambert-Lacroix. Nonparametric confidence set estimation. Mathematical Methods of Statistics , 12(4):410-428, 2003.
[23] Lepski, O.V. On problems of adaptive estimation in white gaussian noise. Topics in Nonparametric Estimation , 12:87-106, 1992. · Zbl 0783.62061
[24] Lepski, O., Nemirovski, A., and Spokoiny, V. On estimation of the \(l_{r}\) norm of a regression function. Probability Theory and Related Fields , 113(2):221-253, 1999. · Zbl 0921.62103
[25] Lepski, O.V., Mammen, E., and Spokoiny, V.G. Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. The Annals of Statistics , 25(3):929-947, 1997. · Zbl 0885.62044
[26] Low, M.G. On nonparametric confidence intervals. The Annals of Statistics , 25(6):2547-2554, 1997. · Zbl 0894.62055
[27] Meyer, Y. Wavelets and Applications. Masson Paris , 1992.
[28] Picard, D. and Tribouley, K. Adaptive confidence interval for pointwise curve estimation. The Annals of Statistics , 28(1):298-335, 2000. · Zbl 1106.62331
[29] Robins, J. and Van Der Vaart, A. Adaptive nonparametric confidence sets. The Annals of Statistics , 34(1):229-253, 2006. · Zbl 1091.62039
[30] Tsybakov, A.B. Introduction à l’estimation non paramétrique. Springer, volume 41, 2004.
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