# Is there a method to estimate distribution parameters given only quantiles?

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?

-
I voted to close this as a duplicate of stats.stackexchange.com/questions/6022, but then it occurred to me that there are possible interpretations of this question that make it different in an interesting way. As a purely mathematical question--if someone teasingly gives you a few quantiles of a mathematical distribution--this is without statistical interest and belongs on the math site. But if these quantiles are measured in a dataset, then generally they will not exactly correspond to the quantiles of any gamma distribution and we need to find the "best" fit in some sense. – whuber Jun 11 '12 at 12:19
So, after that long introductory comment, which situation are you in, Alexx? Should we send your question over to the math people for a theoretical answer, or are these quantiles derived from data? If the latter, then could you help us understand what a "good" (or a "best") solution would look like? E.g., should the fitted distribution match some of the quantiles better than some of the others when a perfect fit is not possible? – whuber Jun 11 '12 at 12:20
But actually the second answer (by @mpiktas) in the link you posted estimates the distribution even if your quantiles are not exact (derived from the data). – Dmitry Laptev Jun 11 '12 at 12:35
@Stas What does this problem have to do with GMM? I don't see any moments in evidence! – whuber Jun 11 '12 at 13:41
"Moments" is a bad name they got stuck with, admittedly. The method in fact works with estimating equations, and I hope you do see some in this example, @whuber. To rephrase, the GMM theory covers anything that can be done with the quadratic loss for estimating equations, including higher order asymptotics and weird dependencies between observations or equations. – StasK Jun 11 '12 at 14:16

i don't know what was in the other post but I have a response. One can look at the order statistics which represent specific quantiles of the distribution namely, the $k$'th order statistic, $X_{(k)}$, is an estimate of the $100 \cdot k/n$'th quantile of the distribution. There is a famous paper in Technometrics 1960 by Shanti Gupta that shows how to estimate the shape parameter of a gamma distribution using the order statistics. See this link: http://www.jstor.org/discover/10.2307/1266548
I TeXed one part of your answer (leaving the content identical) but I'm a little confused and think there may be a typo or something. Re: "One can look at the order statistics which represent specific quantiles of the distribution.....". Do you mean quantiles of the empirical distribution? Also, the $k$'th order statistic usually refers to the $k$'th smallest value, not the $k/n$'th quantile of the empirical distribution, right? Can you clarify (sorry if I'm being dense)? – Macro Jun 11 '12 at 13:11