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Global uniform risk bounds for wavelet deconvolution estimators. (English) Zbl 1209.62060
Summary: We consider the statistical deconvolution problem where one observes $n$ replications from the model $Y=X+\varepsilon$, where $X$ is the unobserved random signal of interest and $\varepsilon $ is an independent random error with distribution $\varphi $. Under weak assumptions on the decay of the Fourier transform of $\varphi $, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators $f_n$ for the density $f$ of $X$, where $f: \Bbb R \rightarrow \Bbb R$ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of $f$ if the Fourier transform of $\varphi $ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if $\varphi $ decays polynomially. We also analyze the case where $f$ is a “supersmooth”/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density $f$.

MSC:
62G07Density estimation
42C40Wavelets and other special systems
62G15Nonparametric tolerance and confidence regions
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References:
[1] Bartlett, P., Boucheron, S. and Lugosi, G. (2002). Model selection and error estimation. Machine Learning 48 85-113. · Zbl 0998.68117 · doi:10.1023/A:1013999503812
[2] Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften 223 . Springer, Berlin. · Zbl 0344.46071
[3] Bissantz, N., Dümbgen, L., Holzmann, H. and Munk, A. (2007). Non-parametric confidence bands in deconvolution density estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 483-506.
[4] Bissantz, N. and Holzmann, H. (2008). Statistical inference for inverse problems. Inverse Problems 24 034009. · Zbl 1137.62325 · doi:10.1088/0266-5611/24/3/034009
[5] Bourdaud, G., Lanza de Cristoforis, M. and Sickel, W. (2006). Superposition operators and functions of bounded p -variation. Rev. Mat. Iberoamericana 22 455-487. · Zbl 1134.46015 · doi:10.4171/RMI/463 · euclid:rmi/116187134 · eudml:41980
[6] Bousquet, O. (2003). Concentration inequalities for sub-additive functions using the entropy method. In Stochastic Inequalities and Applications. Progress in Probability 56 213-247. Birkhäuser, Basel. · Zbl 1037.60015
[7] Butucea, C. and Tsybakov, A. B. (2008a). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 24-39. · Zbl 1141.62021 · doi:10.1137/S0040585X97982840
[8] Butucea, C. and Tsybakov, A. B. (2008b). Sharp optimality in density deconvolution with dominating bias. II. Theory Probab. Appl. 52 237-249. · Zbl 1142.62017 · doi:10.1137/S0040585X97982992
[9] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184-1186. JSTOR: · Zbl 0673.62033 · doi:10.2307/2290153 · http://links.jstor.org/sici?sici=0162-1459%28198812%2983%3A404%3C1184%3AOROCFD%3E2.0.CO%3B2-R&origin=euclid
[10] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004. · Zbl 1137.62323 · doi:10.1088/0266-5611/24/3/034004
[11] Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249-267. · Zbl 05373950
[12] Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523-531. JSTOR: · Zbl 0783.62030 · http://links.jstor.org/sici?sici=0035-9246%281993%2955%3A2%3C523%3AAFATND%3E2.0.CO%3B2-1&origin=euclid
[13] Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1-37. · Zbl 0995.62042 · doi:10.1023/A:1007769924157
[14] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033 · doi:10.1214/aos/1176348248
[15] Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600-610. · Zbl 0785.62038 · doi:10.1214/aos/1176349139
[16] Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907-921. · Zbl 1011.62034 · doi:10.1016/S0246-0203(02)01128-7 · numdam:AIHPB_2002__38_6_907_0 · eudml:77748
[17] Giné, E., Koltchinskii, V. and Sakhanenko, L. (2004). Kernel density estimators: Convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 167-198. · Zbl 1048.62041 · doi:10.1007/s00440-004-0339-x
[18] Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for U -statistics. In High Dimensional Probability, II. Progress in Probability 47 13-38. Birkhäuser, Boston, MA. · Zbl 0969.60024
[19] Giné, E. and Nickl, R. (2009). Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 1605-1646. · Zbl 1255.62103 · doi:10.1214/08-AOP447
[20] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122-1170. · Zbl 1183.62062 · doi:10.1214/09-AOS738
[21] Giné, E. and Nickl, R. (2010). Adaptive estimation of the distribution function and its density by wavelet and spline projections. Bernoulli . · Zbl 1207.62082
[22] Goldenshluger, A. (1999). On pointwise adaptive nonparametric deconvolution. Bernoulli 5 907-925. · Zbl 0953.62033 · doi:10.2307/3318449
[23] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statist. 129 . Springer, New York. · Zbl 0899.62002
[24] Hesse, C. H. and Meister, A. (2004). Optimal iterative density deconvolution. J. Nonparametr. Statist. 16 879-900. · Zbl 1059.62033 · doi:10.1080/10485250410001690086
[25] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547-573. JSTOR: · Zbl 1046.62039 · doi:10.1111/j.1467-9868.2004.02056.x · http://links.jstor.org/sici?sici=1369-7412%282004%2966%3A3%3C547%3AWDIAPS%3E2.0.CO%3B2-X&origin=euclid
[26] Johnstone, I. M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 1781-1804. · Zbl 1056.62044 · doi:10.1214/009053604000000391
[27] Klein, T. and Rio, E. (2005). Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 1060-1077. · Zbl 1066.60023 · doi:10.1214/009117905000000044
[28] Koltchinskii, V. (2001). Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 1902-1914. · Zbl 1008.62614 · doi:10.1109/18.930926
[29] Koltchinskii, V. (2006). Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Statist. 34 2593-2656. · Zbl 1118.62065 · doi:10.1214/009053606000001019
[30] Meister, A. (2008). Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Problems 24 015003. · Zbl 1143.65106 · doi:10.1088/0266-5611/24/1/015003
[31] Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics . Springer, Berlin. · Zbl 1178.62028
[32] Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37 . Cambridge Univ. Press, Cambridge. · Zbl 0776.42019
[33] Nickl, R. and Pötscher, B. M. (2007). Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. J. Theoret. Probab. 20 177-199. · Zbl 1130.46020 · doi:10.1007/s10959-007-0058-1
[34] Nolan, D. and Pollard, D. (1987). U -processes: Rates of convergence. Ann. Statist. 15 780-799. · Zbl 0624.60048 · doi:10.1214/aos/1176350374
[35] Pensky, M. and Sapatinas, T. (2009). Functional deconvolution in a periodic setting: Uniform case. Ann. Statist. 37 73-104. · Zbl 1274.62253 · doi:10.1214/07-AOS552
[36] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053. · Zbl 0962.62030 · doi:10.1214/aos/1017939249
[37] Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229-235. · Zbl 0686.62026 · doi:10.1016/0167-7152(90)90061-B
[38] Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169-184. · Zbl 0697.62035 · doi:10.1080/02331889008802238
[39] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505-563. · Zbl 0893.60001 · doi:10.1007/s002220050108
[40] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation . Springer, Berlin. · Zbl 1176.62032