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

机译:基于强度导数核估计的Spektor-Lord-Willis问题中的渐近置信带

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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.
机译:从线性截面展开球体半径分布的立体问题(称为Spektor-Lord-Willis问题)被表述为Poisson逆问题,并且在某些约束条件下构造了$ L ^ {2} $速率最小极大解Sobolev课程。解决方案是使用边界校正的专用内核类型估计器。首次针对此问题,构建了展开函数的非参数渐近置信带。提出了基于经验风险最小化的自动带宽选择程序。结果表明,带宽选择的Goldenshluger-Lepski过程的一种版本可确保估计器对未知平滑度的适应性。该程序的性能在蒙特卡洛实验中得到证明。

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