Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 144-175.

Confidence bands for multivariate and time dependent inverse regression models

Katharina Proksch, Nicolai Bissantz, and Holger Dette

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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.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 144-175.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597066

Digital Object Identifier
doi:10.3150/13-BEJ563

Mathematical Reviews number (MathSciNet)
MR3322315

Zentralblatt MATH identifier
06436790

Keywords
confidence bands deconvolution inverse problems multivariate regression nonparametric regression rates of convergence time dependent regression function uniform convergence

Citation

Proksch, Katharina; Bissantz, Nicolai; Dette, Holger. Confidence bands for multivariate and time dependent inverse regression models. Bernoulli 21 (2015), no. 1, 144--175. doi:10.3150/13-BEJ563. https://projecteuclid.org/euclid.bj/1426597066


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