I have many different datasets that presumably should follow the same distribution type (but with distinct parameters). I've identified one distribution type that seems to describe best the data (Weibull), and now I've to find the parameters of the distribution for each dataset, and then compare the parameters for each set, so the parameters should be estimated as exactly as possible.
I was using maximum likelihood estimation for estimating the parameters, but noticed (through QQ plots), that for some of the datasets, method of moments works much better (even though for most of the datasets it seems to underperform). I am thinking about estimating the parameters for each dataset through both these methods, and than choose as the final params for the dataset, the more exact ones. What metric/method would you suggest to compare the parameters estimated through Maximum Likelihood Estimation method and the Method of Moments, in order to choose the ones that are better?
Edit: The true parameters are unknown, and and not each dataset follows exactly the specified distribution type, but I need to find the params of a common distribution for each case, that would describe best the data
