New statistical goodness of fit techniques in noisy inhomogeneous inverse problems

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 $\chi^2$ 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 $\chi^2$-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.