# Aternative goodness of fit test when fitting a weibull distribution with parameters estimated from data

I'm an industrial engineering student trying to fit a Weibull distribution to a company's data. The data represents budget overruns and looks like this:

$$\begin{array}{ll} id & amount \\ 1 & 22000 \\ 2 & 28000 \\ 3 & 24000 \\ 4 & 16000 \\ \vdots & \hspace{5mm}\vdots \end{array}$$

My question is twofold. I've read that the KS-test is biased when testing a distribution which has parameters estimated from the data. Since this is my case, I feel like I should account for this in some way. I can find literature on other tests for (log)-normal, exponential and extreme distributions but not for weibull, gamma, beta or student-t. Therefore I'd like to ask if any one can point me in the right direction. Furthermore, if such methods don't exist (or for that matter are too technical for me to implement) what are the consequences for my model, except an extra bias? And can I overcome this if I split my data in 2 sets (1 for fitting and 1 for validation)?

Second (and this is really just for my own informatoin rather than very important for the thesis) I'd like to know what is the best fitting method. I've read that AIC selects the best model in terms of the bias/variance trade-off which I understand. BIC on the other hand is described as "an approximation of a transformation of the bayes factor for a limited set of priors". Which I find hard to interpret. Can anyone provide an (not very technical) explanation?

• Guess I found most of the answers eventually on the web. For future reference of others: see this paper on monte carlo simulation to eventually allow normal KS test: 1 an elaborate explanation here: 2 – Mr. N Aug 2 '17 at 7:01
• (con't) and an implementation in R here: 3 – Mr. N Aug 2 '17 at 7:03