Asymptotics for spectral regularization estimators in statistical inverse problems

While optimal rates of convergence in L2 for spectral regularization estimators in statistical inverse problems have been much studied, the pointwise asymptotics for these estimators have received very little consideration. Here, we briefly discuss asymptotic expressions for bias and variance for some such estimators, mainly in deconvolution-type problems, and also show their asymptotic normality. The main part of the paper consists of a simulation study in which we investigate in detail the pointwise finite sample properties, both for deconvolution and the backward heat equation as well as for a regression model involving the Radon transform. In particular we explore the practical use of undersmoothing in order to achieve the nominal coverage probabilities of the confidence intervals.