Nonparametric statistical inverse problems

Abstract

We explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems. Simple examples are used to present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence, as well as more recent concepts such as adaptive estimation, oracle inequalities, modern model selection methods, Stein's unbiased risk estimation and the very recent risk hull method.

Bayes procedures for adaptive inference in inverse problems for the white noise model

“…Minimax lower bounds for the rate of convergence of estimators for μ are well known in this setting. For instance, the lower bound over Sobolev balls of regularity β > 0 is given by n −β/(1+2β+2 p) and over certain "analytic balls" the lower bound is of the order n −1/2 log 1/2+ p n (see [8]). There are several regularization methods which attain these rates, including classical Tikhonov regularization and Bayes procedures with Gaussian priors.…”

Comparingparameter choice methods for regularization of ill-posed problems

“…For some methods, there are stronger results involving oracle inequalities [19,24,26,25], which, for continuous white noise, have the form…”

“…As for the balancing principle, an estimate of δ 2 ̺ 2 (n) can be obtained from two or more independent sets of data. 25 A c c e p t e d M a n u s c r i p t …”

A two-dimensional backward heat problem with statistical discrete data

“…The main task of the backward problem is of finding the initial temperature from the information of final temperature. As known, the problem is ill-posed (see [13] or Section 3) and, as classified by Cavalier [7], the ill-posedness is severe.…”

“…In our knowledge, we can list here some related papers. Cavalier in [7] gave some theoretical examples about inverse problems with random noise. Mair and Ruymgaart [14] considered theoretical formulas for statistical inverse estimation in Hilbert scales and applied the method for some examples.…”