New statistical goodness of fit techniques in noisy inhomogeneous inverse problems - With application to the recovering of the luminosity distribution of the Milky Way

The assumption that a parametric class of functions fits the data structure sufficiently well is common in fitting curves and surfaces to regression d ata. One then derives a parameter estimate resulting from a least squares f it, say, and in a second step various kinds of chi (2) goodness of fit meas ures, to assess whether the deviation between data and estimated surface is due to random noise and not to systematic departures from the model. In th is paper we show that commonly-used chi (2)-measures are invalid in regress ion models, particularly when inhomogeneous noise is present. Instead we pr esent a bootstrap algorithm which is applicable in problems described by no isy versions of Fredholm integral equations of the first kind. We apply the suggested method to the problem of recovering the luminosity density in th e Milky Way from data of the DIRBE experiment on board the COBE satellite.