Confidence bands for multivariate and time dependent inverse regression models

TL;DR

This paper develops uniform asymptotic confidence bands for multivariate and time-dependent inverse regression functions, using advanced approximation methods, and demonstrates their practical application through simulations and galaxy luminosity data.

Contribution

It introduces the first uniform confidence bands for multivariate nonparametric functions in inverse problems, including time-dependent cases, using strong approximation techniques.

Findings

Confidence bands are valid for multivariate inverse regression functions.

Application to galaxy luminosity profile estimation demonstrates practical utility.

Simulation studies confirm the effectiveness of the proposed methods.

Abstract

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.