XMM-Newton Science Analysis System


bkgfit (ebkgmap-2.10.3) [22.0.0-9173c7d25-20250127]

Goodness of fit:

The likelihood statistic $L$ is just one of many ways of estimating $\chi^2$ for Poissonian data (see [2] for other examples). It is used here in the fitting procedure because it has the advantage that the Hessian components can be calculated analytically. It is however of no use in judging goodness of fit of the final solution, because of a large additive offset between the value of $L$ and the value of $\chi^2$ appropriate to the probability of occurence of any particular $L$ value. A better estimator of $\chi^2$ is the Pearson formula ([2]):

$$\chi^2_{\rm {P}} = \sum_{x=1}^{X}\sum_{y=1}^{Y} \frac{(I_{x,y}-B_{x,y})^2}{B_{x,y}}.$$

This is used in the task to estimate $\chi^2$ and hence a probability that the fitted model $B$ is a good predictor of the data $I$.