Hi Lorenzo, thanks for your feedback and providing me with the theoretical paper.
I think it would be better to explain my situation a bit more detail to get your advice more correctly.
First, I have 650 thousand weighted histograms and I am planning to extract one of the parameters from those histograms and perform the time-resolved analysis. This is my whole analysis plan.
Each histogram contains about 35000 counts, but due to some systematics, the total events in my fit range are approximately 25000 counts. The total number of bins is about 4000 in this fit range. However these events follows exponential decay with respect to particle’s lifetime. Thus, after 2000th bins, there are a few counts in the bin contents. So, basically, if I consider the earlier bins, you’re right. those follow the normal distribution, but I am planning to set the specific fit ranges to get more statistical power and it includes lots of bins including low statistics (the number of counts in each bin are less than 10 after 1500th bins).
Each bin in the histograms are weighted by particular different numbers which are obtained by the error propagation because all the events (counts) are considered identical, independent data.
I have already performed the standard least-square fit and, but I got the biased fit results due to the low statistics. However, when I performed the weighted log likelihood estimation by using the option “WL”, I got more correct central value for the fit parameters and the results were not biased. This is why I started to change the fit method from the chi2 fit to the likelihood method.
As far as I know and I talked with other co-workers, our original counts follows poisson distribution with the different expected counts, but we weighted each bin contents with different values ranging from 0.1 to 0.9. The weights are function of particle’s energy but it is not known.
I really want to confirm how “WL” option is operating and how they calculate the fit uncertainty, and what is the exact form of log likelihood estimation formula, etc.
Would you mind giving me some advice on that?