Electronic Journal of Statistics

Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative

Bogdan Ćmiel, Zbigniew Szkutnik, and Jakub Wojdyła

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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.

Article information

Electron. J. Statist., Volume 12, Number 1 (2018), 194-223.

Received: September 2016
First available in Project Euclid: 8 February 2018

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties 45Q05: Inverse problems

Spektor-Lord-Willis problem inverse problem minimax risk rate of convergence derivative estimation kernel estimator confidence bands adaptive estimator

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Ćmiel, Bogdan; Szkutnik, Zbigniew; Wojdyła, Jakub. Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative. Electron. J. Statist. 12 (2018), no. 1, 194--223. doi:10.1214/18-EJS1391. https://projecteuclid.org/euclid.ejs/1518080460

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