You might think that Poisson as opposed to Gaussian noise would be a better description for the data, but it turns out that by the time you allow for rescaling the noise to cater for extended sky, and weighting the different pixels of the object spectrum in the spatial direction so as to try to maximize the signal-to-noise, the probability distribution function for the signal you are seeking looks horrendously complicated.  Gaussian is simple enough to be easily handled. There is also the complication that rebinning and adding the data on to a uniform wavelength scale, or a logarithmic one (or almost any) introduces correlations between neighboring pixels in the extracted spectrum, and these have to be allowed for in any Chi-squared estimate. We do this by using a scalefactor which is effectively the square root of the error estimate over the actual RMS fluctuations in the spectrum, and applying that on a local basis. This approach has been tested especially by Andrew Cooke, who did not like it, but concluded that it was reasonable after desperately looking for and failing to find an alternative. Having to do this also means that spending a lot of time on error distributions is not very useful.
 

Parameter Error Estimates:

Error estimates in the parameters derived from saturated lines are generally unrealistically small, since they are based on a ellipsoidal approximation to the error contour in which the ellipsoid axes are in the directions of the variables chosen. For saturated lines this is far from being true (see the error contours in the ancient Carswell et al paper on 1101-264 in 1984 ApJ 278, 486). However, I have not done much on single saturated lines with high S/N data except to establish that Ly-a with an estimated logN<14.75 are fairly reliable, and above that the Ly-b provides a better constraint (often by having more than one component).

To get a reasonable error estimate in b/logN for a single line you can use the chi^2 sum technique described by Lampton et al (ApJ 208, 177, 1976 ). You can enter the redshift, Doppler parameter and column density and obtain a chi^2 from the program using the D (or E) options on the first line, and so construct a grid of chi^2 values as a function of the parameters. You can then use the difference of chi^2 from the minimum value to estimate the error ranges (as in Carswell et al, ApJ 278, 486, 1984; Atwood et al, ApJ 292, 58, 1985). This is not desperately efficient, even if you automate it, and it becomes hopeless when attempting to allow for errors in lines that are close together. The ellipsoidal error approximation at least takes reasonable account of these.