Bias-Corrected Confidence Intervals in a Class of Linear Inverse Problems

We propose a new method for constructing confidence intervals in a class of linear inverse problems. Point estimators are obtained via a spectral cutoff method that depends on a regularization parameter α that determines the bias of the estimator. The proposed confidence interval corrects for this bias by explicitly estimating it based on a second regularization parameter ρ that is asymptotically smaller than α. The coverage error of the resulting confidence interval is shown to converge to zero. The proposed method is illustrated by two simulation studies, one in the context of functional linear regression and the other in the context of nonparametric instrumental variables estimation.