Nonparametric bootstrap confidence bands for unfolding sphere size distributions

• Autorzy: Wojdyła Jakub

Identyfikatory

DOI

10.7494/OpMath.2021.41.5.725

• Języki publikacji: EN

Abstrakty

EN

The stereological inverse problem of unfolding the distribution of spheres radii from measured planar sections radii, known as the Wicksell’s corpuscle problem, is considered. The construction of uniform confidence bands based on the smoothed bootstrap in the Wicksell’s problem is presented. Theoretical results on the consistency of the proposed bootstrap procedure are given, where the consistency of the bands means that the coverage probability converges to the nominal level. The finite-sample performance of the proposed method is studied via Monte Carlo simulations and compared with the asymptotic (non-bootstrap) solution described in literature.

Słowa kluczowe

EN

bootstrap; confidence bands; inverse problem; nonparametric density estimation; Wicksell’s problem

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2021
• Tom: Vol. 41, no. 5
• Strony: 725--740
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Wojdyła Jakub

• AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Kraków, Poland

• https://orcid.org/0000-0003-1892-8248

Bibliografia

• [1] A. Antoniadis, J. Fan, I. Gijbels, A wavelet method for unfolding sphere size distributions, The Canadian Journal of Statistics 29 (2001), 251-268.
• [2] P.J. Bickel, M. Rosenblatt, On some global measures of the deviations of density function estimates, The Annals of Statistics 1 (1973), 1071-1095.
• [3] M. Birke, N. Bissantz, H. Holzmann, Confidence bands for inverse regression models, Inverse Problems 26 (2010), Article 115020.
• [4] N. Bissantz, L. Dümbgen, H. Holzmann, A. Munk, Non-parametric confidence bands in deconvolution density estimation, Journal of the Royal Statistical Society: Series B 69 (2007), 483-506.
• [5] N. Bissantz, H. Holzmann, Statistical inference for inverse problems, Inverse Problems 24 (2008), Article 034009.
• [6] N. Bissantz, H. Holzmann, K. Proksch, Confidence regions for images observed under the Radon transform, Journal of Multivariate Analysis 128 (2014), 86-107.
• [7] S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, Wiley, Chichester, 2013.
• [8] L.M. Cruz-Orive, Distribution-free estimation of sphere size distributions from slabs showing overprojections and truncations, with a review of previous methods, J. Microscopy 131 (1983), 265-290.
• [9] M. Csörgo, P. Revesz, Strong Approximations in Probability and Statistics, Akademiai Kiadó, Budapeszt, 1981.
• [10] B. ćmiel, Z. Szkutnik, J. Wojdyła, Asymptotic confidence bands in the Spektor- -Lord-Willis problem via kernel estimation of intensity derivative, Electronic Journal of Statistics 12 (2018), 194-223.
• [11] D. De Angelis, G.A. Young, Smoothing the bootstrap, International Statistical Review 60 (1992), 45-56.
• [12] A. Delaigle, P. Hall, F. Jamshidi, Confidence bands in nonparametric errors-in-variables regression, Journal of the Royal Statistical Society: Series B 77 (2015), 149-169.
• [13] B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics 7 (1979), 1-26.
• [14] B. Efron, The Jackknife, the Bootstrap and Other Resarnpling Plans, SIAM, Philadelphia, 1982.
• [15] B. Efron, G. Gong, A leisurely look at the bootstrap, the jackknife and crossvalidation, The American Statistician 37 (1983), 36-48.
• [16] E. Gine, R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge University Press, New York, 2016.
• [17] G.K. Golubev, B.Y. Levit, Asymptotical ly efficient estimation in the Wicksel l problem, Annals of Statistics 26 (1998), 2407-2419.
• [18] P. Groeneboom, G. Jongbloed, Nonparametric Estimation under Shape Constraints, Cambridge University Press, New York, 2014.
• [19] P. Hall, T.J. DiCiccio, J.P. Romano, On smoothing and the bootstrap, Annals of Statistics 17 (1989), 692-704.
• [20] P. Hall, R.L. Smith, The kernel method for unfolding sphere size distributions, Journal of Computational Physics 74 (1988), 409-421.
• [21] K. Kato, Y. Sasaki, Uniform confidence bands in deconvolution with unknown error distribution, Journal of Econometrics 207 (2018), 129-161.
• [22] K. Kato, Y. Sasaki, Uniform confidence bands for nonparametric errors-in-variables regression, Journal of Econometrics 213 (2019), 516-555.
• [23] K. Lounici, R. Nickl, Global uniform risk bounds for wavelet deconvolution estimators, Annals of Statistics 39 (2011), 201-231.
• [24] K. Proksch, N. Bissantz, H. Dette, Confidence bands for multivariate and time dependent inverse regression models, Bernoulli 21 (2015), 144-175.
• [25] B.W. Silverman, G.A. Young, The bootstrap: to smooth or not to smooth?, Biometrika 74 (1987), 469-479.
• [26] W. Stute, A law of the logarithm for kernel density estimators, The Annals of Probability 10 (1982), 414-422.
• [27] C.C. Taylor, A new method for unfolding sphere size distributions, Journal of Microscopy 132 (1983), 57-66.
• [28] S.D. Wicksell, The corpuscle problem: A mathematical study of a biometric problem, Biometrika 17 (1925), 84-99.
• [29] B.B. Winter, Convergence rate of perturbed empirical distribution functions, Journal of Applied Probability 16 (1979), 163-173.
• [30] J. Wojdyła, Z. Szkutnik, Nonparametric confidence bands in Wicksel l’s problem, Statis- tica Sinica 28 (2018), 93—113.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).