Minimax theory for a class of nonlinear statistical inverse problems

TY - JOUR

T1 - Minimax theory for a class of nonlinear statistical inverse problems

AU - Ray, Kolyan

AU - Schmidt-Hieber, Johannes

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - nonlinear statistical inverse problems

KW - adaptive estimation

KW - nonparametric regression

KW - thresholding

KW - wavelets

U2 - 10.1088/0266-5611/32/6/065003

DO - 10.1088/0266-5611/32/6/065003

M3 - Article

SN - 0266-5611

JO - Inverse Problems

JF - Inverse Problems

ER -