9
$\begingroup$

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:

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

  2. 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.

$\endgroup$
  • $\begingroup$ How do you define "well shaped" ? $\endgroup$ – Tim Apr 19 '18 at 7:31
  • $\begingroup$ @Tim It looks like gamma-distribution. I've edited the question accordingly. $\endgroup$ – zlon Apr 19 '18 at 7:43
  • 1
    $\begingroup$ 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. $\endgroup$ – Xi'an Apr 19 '18 at 9:20
  • 1
    $\begingroup$ I don't need the unique distribution. I need ANY with given moments. $\endgroup$ – zlon Apr 19 '18 at 9:21
  • 2
    $\begingroup$ If any solution suffices, use your data. $\endgroup$ – Nick Cox Apr 19 '18 at 11:35
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.