# Recovering Distribution from Percentiles

I have data on the 10th, 25th, 50th, 75th, and 90th percentiles of a probability distribution, together with the mean, and standard deviation. I am interested in recovering a continuous distribution that would match or approximate well these data points with a flexible family of distributions.

One possibility is to let the density be flat between the percentiles and choose the 0th and 100th percentiles to match the mean and variance. This procedure, unfortunately, does not lead to reasonable results.

I have to do this many times, but just for concreteness here is one example:

$$p_{10}=-0.89, \quad p_{25}= -0.20, \quad p_{50}= 0.08, \quad p_{75}= 0.33, \quad p_{90}= 0.71,$$

with

$$\text{mean} = -0.21 \quad\text{and}\quad\text{std dev}=2.25.$$

I have tried the Generalized normal distribution, and the Exponentially modified Gaussian distribution, but they do not seem to be able to approximate the percentiles well enough.

I have also tried the method proposed in the answer to this question using many different distributions besides the standard normal, this allows to approximate the percentiles well, but then the standard deviation is always underestimated.

Any suggestions would be very welcome!

• Can the 0% and 100% percentiles be $-\infty$ and $\infty$ respectively? I wouldn't try to rebuild those percentiles,as many well-known distributions behaive this way – David Jul 4 '19 at 7:28
• I am afraid that little amount of data will give you not too much information unless you restrict your choices to a very particular set of possible distributions – David Jul 4 '19 at 7:30
• Yes, in principle the support could be the real line, but I think it would be sensible to impose bounds if that is going to help to match the moments. I am not sure I understand what you mean with your second comment, could you elaborate? – mzp Jul 4 '19 at 13:30
• But the 0% and 100% percentiles are often something hard to estimate. With my second comment, I mean that, unless you already have a pretty clear idea of what distribution family the data could be, you won't be able to guess it only from such little information – David Jul 4 '19 at 13:38
• My prior was the same: many distributions should be able to deliver these numbers, so they don't identify the distribution. But my prior has shifted substantially, I've tried pretty much all suitable distributions I could find and none are able to get close. I either get the percentiles or the standard deviation right, never both to a reasonable degree. – mzp Jul 4 '19 at 13:42