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Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative. (English) Zbl 1387.62061
Summary: The stereological problem of unfolding the distribution of spheres radii from linear sections, known as the Spektor-Lord-Willis problem, is formulated as a Poisson inverse problem and an \(L^2\)-rate-minimax solution is constructed over some restricted Sobolev classes. The solution is a specialized kernel-type estimator with boundary correction. For the first time for this problem, non-parametric, asymptotic confidence bands for the unfolded function are constructed. Automatic bandwidth selection procedures based on empirical risk minimization are proposed. It is shown that a version of the Goldenshluger-Lepski procedure of bandwidth selection ensures adaptivity of the estimators to the unknown smoothness. The performance of the procedures is demonstrated in a Monte Carlo experiment.

62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
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[1] Antoniadis, A. and Bigot, J. (2006). Poisson inverse problems., The Annals of Statistics 34 2132-2158. · Zbl 1106.62035
[2] Barthel, M., Klimanek, P. and Stoyan, D. (1985). Stereological substructure analysis in hot-deformed metals from TEM-images., Czech. Journal of Physics B 35 265-268.
[3] Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates., The Annals of Statistics 1 1071-1095. · Zbl 0275.62033
[4] Birke, M., Bissantz, N. and Holzmann, H. (2010). Confidence bands for inverse regression models., Inverse Problems 26 article 115020. · Zbl 1203.62060
[5] Bissantz, N. and Birke, M. (2009). Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators., Journal of Multivariate Analysis 100 2364-2375. · Zbl 1175.62035
[6] Bissantz, N., Dümbgen, L., Holzmann, H. and Munk, A. (2007). Non-parametric confidence bands in deconvolution density estimation., Journal of the Royal Statistical Society, Ser. B 69 483-506.
[7] Bissantz, N. and Holzmann, H. (2008). Statistical inference for inverse problems., Inverse Problems 24 article 034009. · Zbl 1137.62325
[8] Bissantz, N., Holzmann, H. and Proksch, K. (2014). Confidence regions for images observed under the Radon transform., Journal of Multivariate Analysis 128 86-107. · Zbl 1291.65369
[9] Chernozhukov, V., Chetverikov, D. and Kato, K. (2014). Anti-concentration and honest, adaptive confidence bands., The Annals of Statistics 42 1787-1818. · Zbl 1305.62161
[10] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013)., Stochastic Geometry and its Applications. Wiley, Chichester. · Zbl 1291.60005
[11] Ćmiel, B. (2010). An adaptive wavelet shrinkage approach to the Spektor-Lord-Willis problem., Journal of Multivariate Analysis 101 1458-1470. · Zbl 1190.62153
[12] Ćmiel, B. (2012). Poisson intensity estimation for the Spektor-Lord-Willis problem using a wavelet shrinkage approach., Journal of Multivariate Analysis 112 194-206. · Zbl 1273.62068
[13] Delaigle, A., Hall, P. and Jamshidi, F. (2015). Confidence bands in nonparametric errors-in-variables regression., Journal of the Royal Statistical Society, Ser. B 77 149-169. · Zbl 1414.62129
[14] Dony, J. and Einmahl, U. (2006). Weighted uniform consistency of kernel density estimators with general bandwidth sequences., Electronic Journal of Probability 11 844-859. · Zbl 1107.62030
[15] Doumic, M., Hoffmann, M., Reynaud-Bouret, P. and Rivoirard, V. (2012). Nonparametric estimation of the division rate of a size-structured population., SIAM Journal on Numerical Analysis 50 925-950. · Zbl 1317.92063
[16] Dudek, A. and Szkutnik, Z. (2008). Minimax unfolding of the spheres size distribution from linear sections., Statistica Sinica 18 1063-1080. · Zbl 1149.62074
[17] Eubank, R. L. and Speckman, P. L. (1993). Confidence bands in nonparametric regression., Journal of the American Statistical Association 88 1287-1301. · Zbl 0792.62030
[18] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation., The Annals of Statistics 38 1122-1170. · Zbl 1183.62062
[19] Giné, E. and Nickl, R. (2016)., Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York.
[20] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality., The Annals of Statistics 39 1608-1632. · Zbl 1234.62035
[21] Han, J. H. and Kim, D. Y. (1998). Determination of three-dimensional grain size distribution by linear intercept measurement., Acta Materialia 46 2021-2028.
[22] Hilliard, J. E. and Lawson, L. R. (2003)., Stereology and Stochastic Geometry. Kluwer, Dordrecht. · Zbl 1108.60008
[23] Lord, G. W. and Willis, T. F. (1951). Calculation of air bubble distribution from results of a Rosiwal traverse of aerated concrete., A.S.T.M. Bulletin 56 177-187.
[24] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimator., The Annals of Statistics 39 201-231. · Zbl 1209.62060
[25] Proksch, K., Bissantz, N. and Dette, H. (2015). Confidence bands for multivariate and time dependent inverse regression models., Bernoulli 21 144-175. · Zbl 1388.62113
[26] Reiss, R. D. (1993)., A Course on Point Processes. Springer, New York. · Zbl 0771.60037
[27] Reynaud-Bouret, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities., Probability Theory and Related Fields 126 103-153. · Zbl 1019.62079
[28] Reynaud-Bouret, P., Rivoirard, V., Grammont, F. and Tuleau-Malot, C. (2014). Goodness-of-fit tests and nonparametric adaptive estimation for spike train analysis., Journal of Mathematical Neuroscience 4:3 1-41. · Zbl 1321.92047
[29] Silverman, B. W. (1986)., Density Estimation for Statistics and Data Analysis. Chapman and Hall, London. · Zbl 0617.62042
[30] Spektor, A. G. (1950). Analysis of distribution of spherical particles in non-transparent structures., Zavodskaja Laboratorija 16 173-177.
[31] Szkutnik, Z. (2000). Unfolding intensity function of a Poisson process in models with approximately specified folding operator., Metrika 52 1-26. · Zbl 1093.62570
[32] Szkutnik, Z. (2005). B-splines and discretization in an inverse problem for Poisson processes., Journal of Multivariate Analysis 93 198-221. · Zbl 1087.62094
[33] Szkutnik, Z. (2007). Unfolding spheres size distribution from linear sections with B-splines and EMDS algorithm., Opuscula Mathematica 27 151-165. · Zbl 1145.60009
[34] Szkutnik, Z. (2010). A note on minimax rates of convergence in the Spektor-Lord-Willis problem., Opuscula Mathematica 30 203-207. · Zbl 1222.62105
[35] Tsybakov, A. B. (2009)., Introduction to Nonparametric Estimation. Springer, New York. · Zbl 1176.62032
[36] Wojdyła, J. and Szkutnik, Z. (2018). Nonparametric confidence bands in Wicksell’s problem., Statistica Sinica 28 93-113. · Zbl 1382.62022
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