Here, y is the number of quantum defects observed in iodine atoms and x is the excited energy state. As a software test, this example is considered of average difficulty in spite of the fact that there are a number of local optima very close to the global optimum shown in the figure.
The indicated optimization was carried out using unweighted least-squares [R-squared = 0.99841]. This assumes that the residuals (deviations from the curve along the y-axis) are Gaussian, with zero mean and the same variance at all points. However, residuals are often not Gaussian but Laplacian (double-exponential). If this is the case, then the least-squares criterion is invalid and the ML parameters must be found by minimizing the average magnitude of the deviations (residuals). Standard regression packages do not offer this option but Regress+ does. When this new criterion is employed, we get new ML parameters, as follows:
With the minimum average deviation criterion, R-squared decreases to 0.99838 but the average deviation is 3.24272e-03 whereas it was 3.28547e-03 with the least-squares criterion.
Needless to say, Regress+ is ideally suited to assessing whether or not a set of residuals is Gaussian or Laplacian (or something else!) As it happens, the residuals from this least-squares fit are acceptably Gaussian and acceptably Laplacian, probably because the dataset is not big enough to show the difference.