is there a way to fit a specified distribution if you are only given a few quantiles?
For example, if I told you I have a gamma distributed data set, and the empirical 20%, 30%, 50% and 90%-quantiles are, respectively:
20% 30% 50% 90% 0.3936833 0.4890963 0.6751703 1.3404074
How would I go and estimate the parameters? Are there multiple ways to do that, or is there already a specific procedure?
more edit: I don't specifically ask for the gamma distribution, this was just an example because I worry I can't explain my question appropriately. My task is that I have some (2-4) given quantiles, and want to estimate the (1-3) parameters of a few distributions as "close" as possible. Sometimes there's an (or infinite) exact solution(s), sometimes not, right?