# 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, 2012 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, 2012 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). Jun 11, 2012 at 12:35
• @Stas What does this problem have to do with GMM? I don't see any moments in evidence!
– whuber
Jun 11, 2012 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. Jun 11, 2012 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)? Jun 11, 2012 at 13:11