New statistical goodness of fit techniques in noisy inhomogeneous inverse problems

With application to the recovering of the luminosity distribution of the Milky Way

¹ Astronomisches Institut der Universität Basel, Venusstr. 7, 4102 Binningen/Basel, Switzerland
² Fakultät für Mathematik und Informatik der Universität GH Paderborn, Warburgerstr. 100, 33098 Paderborn, Germany

Corresponding author: N. Bissantz, bissantz@astro.unibas.ch

Received: 16 June 2000
Accepted: 12 June 2001

Abstract

The assumption that a parametric class of functions fits the data structure sufficiently well is common in fitting curves and surfaces to regression data. One then derives a parameter estimate resulting from a least squares fit, say, and in a second step various kinds of goodness of fit measures, to assess whether the deviation between data and estimated surface is due to random noise and not to systematic departures from the model. In this paper we show that commonly-used -measures are invalid in regression models, particularly when inhomogeneous noise is present. Instead we present a bootstrap algorithm which is applicable in problems described by noisy versions of Fredholm integral equations of the first kind. We apply the suggested method to the problem of recovering the luminosity density in the Milky Way from data of the DIRBE experiment on board the COBE satellite.