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Minimax theory for a class of non-linear statistical inverse problems

Research paper by Kolyan Ray, Johannes Schmidt-Hieber

Indexed on: 11 May '16Published on: 11 May '16Published in: Mathematics - Statistics



Abstract

We study a class of statistical inverse problems with non-linear pointwise operators motivated by concrete statistical applications. A two-step procedure is proposed, where the first step smoothes the data and inverts the non-linearity. This reduces the initial non-linear 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 H\"older smoothness scales that are natural in this setting.