Single model for a different data set I have a single model (e.g generalized Pareto distribution) to test with a different data set (I have a set of different increasing threshold and fit the same model with a data above these threshold). I want to know which set of data will give me the best model fit. Can i use likelihood ratio test in this case or any other suggestion to get this? Many thanks in advance. 
 A: You can't use the standard likelihood ratio test. It only works for comparing the likelihoods of different models on the same dataset.
For the more general question of how to pick the set to which the model fits best, you need to define what "best" means. If your data is iid, the likelihood is obtained by multiplying a probability value (a number less than one) for each point. Just comparing likelihood will favor small sets (because fewer small numbers are multiplied). This is probably not what you want. 
GP has two parameters, $x_m$ and $\alpha$. I assume you fit both of them on each dataset (because if you hold $x_m$ fixed and change the threshold, then for larger thresholds there won't be any data near $x_m$). In this case, the best fit will be when your threshold is such that there is only one point left. But this fit only looks best because the problem of fitting one data point is very easy. Fitting many points is more difficult, so naturally the fit looks worse. So you need to account for the difficulty of the problem somehow. The problem seems complementary to the usual problem of accounting for the "model complexity". I'm not sure whether there is a standard way of doing this. This would certainly be interesting to learn about.
Meanwhile, a way to sidestep this would be the following. Do you have a model for what the data below the threshold should look like? Maybe they come from a different process for which you know the distribution. If yes, what you can do is fit a mixture model to the data both above and below threshold. This mixture would always fit the same dataset (namely, all of your points). So for the mixture you could use the likelihood test or some other model selection method.
