Distribution fitting on empirical PDF? I have a class of distributions that I think may have generated some data I have, and would like to find the best fit distribution and corresponding parameters. But the data I have aren't samples from a distribution, they are an empirical PDF. 
 A: Do you have some knowledge on how the empirical PDF was obtained ?
Like if it was obtained from an histogram with entries / bin, then you can perform a max likelihood fit or a least square fit of your class of distributions directly on the histogram data.
The fit setup depends on your histogram:


*

*if bins have small counts (like < 20 entries), it is better to describe each bin by a Poisson (when N is not fixed) or a multinomial (when N is fixed) PDF and then perform a maximum likelihood fit.

*if bins have large counts, then you can approximate the PDF of each bin as a normal distribution and perform a least square fit with sqrt(entries) as bin uncertainty. The advantage of this approach is that you can quantify the goodness-of-fit via the resulting chi2Min after fit.


HTH
A: If you have a vectorof values and probabilities, one simple way to fit a distribution is to create a "theoretical" vector along side the empirical one, and use a non-linear solver (SOLVER in Excel if the distribution has 2 or fewer parameters, nloptr or optim or nlminb in R, etc.) to minimize the sum of squared differences between the empirical and the test vector.
