Use results from two-sample Kolmogorov-Smirnov to compare methods Let say I observe a phenomenon an I get its PDF. Then I come up with two models to simulate this phenomenon. From these models, I can get a PDF and I want to know which model is better.
I am not interested in the $Q_2$ of the model. Here the objective is to get the PDF to assess threashold exceeding or to see how the phenomenon responds to uncertainties for instance. Maybe there is a bi-modal structure, particular values with particular level of probabilities, etc. This is for exploratory so no particular quantile is targeted and thus I need the PDF from the model which represents best the PDF from the real observations.
So let’s say I have:


*

*One sample from observation giving PDF1,

*One sample from the first model giving PDF2,

*One sample from the second model giving PDF3.


I am not a statistician, so from my findings I have to use a Kolmogorov-Smirnov test to assess if two PDFs can be considered "equal".
Thus, I compute two-sample Kolmogorov-Smirnov test with PDF1 vs PDF2 giving pvalue1 and from PDF1 vs PDF3 giving pvalue2. 
Can I compare the two pvalues? If pvalue1 < pvalue2, can I say that the first model is better?
Or maybe this approach is wrong, so how to tell which model is better at getting the PDF (only the PDF, I know about Q2 and this is not what I am looking for)?
 A: A two-sample Kolmogorov-Smirnov test comparing data to values simulated from a model is not a good way to see whether a distributional model fits the data.
In the case where you have not estimated any parameters, it's simply a much less efficient way of doing a one-sample K-S test. You can improve its efficiency by drawing a larger simulation sample. As the sample size goes to infinity, guess what you end up with? The one-sample K-S test!
Neither test is suitable (as is) if you estimate parameters, though.
One of the things you're after is to be able to estimate the probability to exceed a threshold. One way to get it is to try to identify a pdf, but it's often not the best way to achieve that goal  (in part it depends on how far into the tail the threshold is). If the thresholds are near the middle, you may, for example, be better to estimate these directly from the data.
if those threshholds are not mostly near the middle, it may make sense try to identify a pdf for that purpose, but then I probably wouldn't be using K-S distance to assess fit, but something that would do better in the tail. 
(You can also get a good idea how a system behaves without identifying a distributional form for data, since you have the ECDF itself)
However; if you need information about the extreme tail (beyond almost all the data) then the data isn't much use; you need some assumptions as well.
Now lastly, none of this is necessarily of the greatest use for deciding between two competing models.
