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Uniform asymptotic confidence bands for a multivariate regression function inan inverse regression model with a convolution-type operator are constructed.The results are derived using strong approximation methods and a limit theoremfor the supremum of a stationary Gaussian field over an increasing system ofsets. As a particular application, asymptotic confidence bands for a timedependent regression function $f_t(x)$ ($x\in \mathbb {R}^d,t\in \mathbb {R}$)in a convolution-type inverse regression model are obtained. Finally, wedemonstrate the practical feasibility of our proposed methods in a simulationstudy and an application to the estimation of the luminosity profile of theelliptical galaxy NGC5017. To the best knowledge of the authors, the resultspresented in this paper are the first which provide uniform confidence bandsfor multivariate nonparametric function estimation in inverse problems.

Confidence bands for multivariate and time dependent inverse regression models