A note on minimax rates of convergence in the Spektor-Lord-Willis problem

• Autorzy: Szkutnik Z.
• Języki publikacji: EN

Abstrakty

EN

In this note, attainable lower bounds are constructed for the convergence rates in a stereological problem of unfolding spheres size distribution from linear sections, which shows that a spectral type estimator is strictly rate minimax over some Sobolev-type classes of functions.

Słowa kluczowe

EN

poisson inverse problem; rate minimaxity; singular value decomposition; stereology

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2010
• Tom: Vol. 30, no. 2
• Strony: 203--207
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Szkutnik Z.

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

Bibliografia

• [1] L. Birge, The Brouwer Lecture 2005. Statistical estimation with model selection, 2006. (available at arXiv:math.ST/0605187 v2)
• [2] A. Dudek, Z. Szkutnik, Minimax unfolding spheres size distribution from linear sections, Statist. Sinica 18 (2008), 1063–1080.
• [3] J.-H. Han, D.-Y. Kim, Determination of three-dimensional grain size distribution by linear intercept measurement, Acta Mater. 46 (1998), 2021–2028.
• [4] J.E. Hilliard, L.R. Lawson, Stereology and Stochastic Geometry, Kluwer, Dordrecht, 2003.
• [5] I.M. Johnstone, B.W. Silverman, Speed of estimation in positron emission tomography and related inverse problems, Ann. Statist. 18 (1990), 251–280.
• [6] G.W. Lord, T.F. Willis, Calculation of air bubble distribution from results of a Rosiwal traverse of aerated concrete, A.S.T.M. Bull. 56 (1951), 177–187.
• [7] A.C.M. van Rooij, F.H. Ruymgaart, Asymptotic minimax rates for abstract linear estimators, J. Statist. Plann. Inference 53 (1996), 389-402.
• [8] A.G. Spektor, Analysis of distribution of spherical particles in non-transparent structures, Zavodskaja Laboratorija 16 (1950), 173–177.
• [9] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, Akademie-Verlag, Berlin, 1987.
• [10] Z. Szkutnik, A new solution to an old stereological problem, Bulletin of the International Statistical Institute LXII (2007), 4577–4580.