Confidence bands for multivariate and time dependent inverse regression models

Uniform asymptotic confidence bands for a multivariate regression function in an in-verse regression model with a convolution-type operator are constructed. The results arederived using strong approximation methods and a limit theorem for the supremum of astationary Gaussian field over an increasing system of sets. As a particular applicationasymptotic confidence bands for a time dependent regression function ft(x) (x ∈ Rd, t ∈ R)in a convolution-type inverse regression model are obtained. To the best knowledge of theauthors the results presented in this paper are the first which provide uniform confidencebands for multivariate nonparametric function estimation in inverse problems.