Generate random variable with given moments

Question

I know first $N$ moments of some distribution. I also know that my distribution is continuous, unimodal and well shaped (it looks like gamma-distribution). Is it possible to:

Using some algorithm, generate samples from this distribution, which in limit conditions will have exactly the same moments?
Solve this problem analytically?

I understand that until I have an infinite number of moments, this question cannot have a unique solution. I would be happy to have any.

Due to the comments clarification: I don't need to restore original distribution. I need ANY with a given moments.

@Tim It looks like gamma-distribution. I've edited the question accordingly. — zlon, Apr 19 '18 at 07:43
Can you give some context? Is this moments given theoretically or empirically? — kjetil b halvorsen, Apr 19 '18 at 08:58
I have some numbers, but, as I understand, the group who provide me these numbers has an analytical solution for any given moment, but not for distribution itself. — zlon, Apr 19 '18 at 09:06
You cannot generate from a distribution for which you only know moments. There even exist cases where the whole sequence of moments is not sufficient to specify the distribution uniquely. — Xi'an, Apr 19 '18 at 09:20
I don't need the unique distribution. I need ANY with given moments. — zlon, Apr 19 '18 at 09:21
For the moments of a normal distribution it is trivial. But otherwise? Very good question. — Michael M, Apr 19 '18 at 09:43
As I understand normal distribution is fully determined by first 2 moments. — zlon, Apr 19 '18 at 09:54
But, the numbers that you have, are they exact, theretically given numbers, or estimates from data? — kjetil b halvorsen, Apr 19 '18 at 10:56
They are exact. So, I need to find ANY analytical solution, of numerical solution, with prove, that in limit case i will have given moments — zlon, Apr 19 '18 at 10:59
https://stats.stackexchange.com/questions/141652/constructing-a-continuous-distribution-to-match-m-moments?rq=1 — kjetil b halvorsen, Jan 02 '21 at 15:36

score 6 · Accepted Answer · edited Apr 19 '18 at 12:25

We really need that you give some more information as asked for in comments.

There is a monograph https://projecteuclid.org/euclid.lnms/1249305333 dedicted to your question.

Here: http://fks.sk/~juro/docs/paper_spie_2.pdf is another paper.

Some related posts on sister sites:

https://math.stackexchange.com/questions/386025/finding-a-probability-distribution-given-the-moment-generating-function

https://mathoverflow.net/questions/3525/when-are-probability-distributions-completely-determined-by-their-moments

Another paper is http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.106.6130 Its author lists some possible approaches, like maximum entropy methods (Jaynes 1994), a method of obtaining upper and lower bounds on the cumulative distribution function (cdf) using the first $n$ moments (https://www.semanticscholar.org/paper/A-moments-based-distribution-bounding-method-R%C3%A1cz-Tari/cd28087b8ead5c4d5c4eebc2b91e2a4b8caef3f3), but then diced to assume a unimodal distribution and fit to a flexible distribution family, like Pearson family, Johnson family or Generalized Tukey Lambda family. Finally she implements a solution based on fitting first four moments to the Generalized Lambda family.

Generate random variable with given moments

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