Minimax theory for a class of nonlinear statistical inverse problems. (25th April 2016)

• Record Type: Journal Article
• Authors: Ray, Kolyan; Schmidt-Hieber, Johannes
• Is Part Of: Inverse problems. Volume 32:Number 6(2016:Jun.)
• Publication Date: 2016-04-25
• Subjects: nonlinear statistical inverse problems -- adaptive estimation -- nonparametric regression -- thresholding -- wavelets; primary 62G05 -- secondary 62G08 -- 62G20; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/6/065003
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We study a class of statistical inverse problems with nonlinear pointwise operators motivated by concrete statistical applications. A two-step procedure is proposed, where the first step smoothes the data and inverts the nonlinearity. This reduces the initial nonlinear problem to a linear inverse problem with deterministic noise, which is then solved in a second step. The noise reduction step is based on wavelet thresholding and is shown to be minimax optimal (up to logarithmic factors) in a pointwise function-dependent sense. Our analysis is based on a modified notion of Hölder smoothness scales that are natural in this setting.