Confidence bands for multivariate and time dependent inverse regression models.(English)Zbl 1388.62113

Summary: Uniform asymptotic confidence bands for a multivariate regression function in an inverse regression model with a convolution-type operator are constructed. The results are derived using strong approximation methods and a limit theorem for the supremum of a stationary Gaussian field over an increasing system of sets. As a particular application, asymptotic confidence bands for a time dependent regression function $$f_{t}(x)$$ ($$x\in\mathbb{R} ^{d}$$, $$t\in\mathbb{R}$$) in a convolution-type inverse regression model are obtained. Finally, we demonstrate the practical feasibility of our proposed methods in a simulation study and an application to the estimation of the luminosity profile of the elliptical galaxy NGC5017. To the best knowledge of the authors, the results presented in this paper are the first which provide uniform confidence bands for multivariate nonparametric function estimation in inverse problems.

MSC:

 62G08 Nonparametric regression and quantile regression 62G07 Density estimation 62G15 Nonparametric tolerance and confidence regions

GAUSSIAN
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