A graphical selection method for parametric models in noisy inhomogeneous regression

A common problem in physics is to fit regression data by a parametric class of functions, and to decide whether a certain functional form allows for a good fit of the data. Common goodness of fit methods are based on the calculation of the distribution of certain statistical quantities under the assumption that the model under consideration holds true. This proceeding bears methodological flaws, e.g. a good “fit” albeit the model is wrong might be due to over-fitting, or to the fact that the chosen statistical criterion is not powerful enough against the present particular deviation between model and true regression function. This causes particular difficulties when models with different numbers of parameters are to be compared. Therefore the number of parameters is often penalised additionally. We provide a methodology which circumvents these problems to some extent. It is based on the consideration of the error distribution of the goodness of fit criterion under a broad range of possible models and not only under the assumption that a given model holds true. We present a graphical method to decide for the most evident model from a range of parametric models of the data. The method allows to quantify statistical evidence for the model (up to some distance between model and true regression function) and not only absence of evidence against, as common goodness of fit methods do. Finally we apply our method to the problem of recovering the luminosity density of the Milky Way from a de-reddened COBE/DIRBE L-band map. We present statistical evidence for flaring of the stellar disc inside the solar circle.