Uncertainty Limits on Solutions of Inverse Problems over Multiple Orders of Magnitude using Bootstrap Methods: An Astroparticle Physics Example

Abstract

Astroparticle experiments such as IceCube or MAGIC require a deconvolution of their measured data with respect to the response function of the detector to provide the distributions of interest, e.g. energy spectra. In this paper, appropriate uncertainty limits that also allow to draw conclusions on the geometric shape of the underlying distribution are determined using bootstrap methods, which are frequently applied in statistical applications. Bootstrap is a collective term for resampling methods that can be employed to approximate unknown probability distributions or features thereof. A clear advantage of bootstrap methods is their wide range of applicability. For instance, they yield reliable results, even if the usual normality assumption is violated. The use, meaning and construction of uncertainty limits to any user-specific confidence level in the form of confidence intervals and levels are discussed. The precise algorithms for the implementation of these methods, applicable for any deconvolution algorithm, are given. The proposed methods are applied to Monte Carlo simulations to show their feasibility and their precision in comparison to the statistical uncertainties calculated with the deconvolution software TRUEE.