Goodness of Chi-Square for Linearly Parameterized Fitting

The paper shows an alternative perspective of the reduced chi-square as a measure of the goodness of fitting methods. The reduced chi-square is given by the ratio of the fitting over the propagation errors, that is, a universal relationship that holds for any linearity, but not for a nonlinearly parameterized fitting model. We begin by providing the proof for the traditional examples of one-parametric fitting of a constant and the bi-parametric fitting of a linear model, and then, for the general case of any linearly multi-parameterized model. We also show that this characterization is not generally true for nonlinearly parameterized fitting. Finally, we demonstrate these theoretical developments with an application in real data from the plasma protons in the heliosphere.