# Generation of a multivariate distribution, given its marginal moments and correlation matrix

I'd like to draw from a multivariate distribution (of any type), conditional to knowing only the moments of the the marginal distributions and the overall covariance (or correlation) matrix. Clearly, I'd like the drawn multivariate distributions to preserve the given moment characteristics of each marginal distribution, as well as the given overall correlation.

I could get some hints here, and here, which use Cholesky decomposition of the covariance matrix. Also I found suggestion here, and here, which use a sort of inverse probability sampling approach.

Both methods can be envisioned as transformation from (multi)-normal variates. In particular, being the Cholesky-decomposition method a linear transformation, I get the feeling that it applies well when generating multivariate normals, but performs poorly in other non-normal cases. For instance, the first question is

Q1) How to incorporate information on third and fourth marginal moments, while applying the Cholesky transform, such that the resulting multivariate draw does keep marginal moments invariant ? (first and second marginal moments can be already made transformation-invariant, as shown in the links above).

The second approach based on inverse sampling seems an elegant one, although, there too, departure from normality in the simulated data can yield marginal moments or correlation structure which are different from the one given. Here too, would be nice to know if the approach can be made general, such that any type of multivariate distribution can be generated.

Thus, the second and generic question is

Q2) Does anyone know a method to draw from a multivariate distribution, only given the first four moments of each marginal distribution and overall correlation matrix, and such that both marginal moments and correlation matrix of the drawn data do match the ones given ?

Thank for any eventual help.

## 2 Answers

In the paper (Hoyland et al, 2000) an algorithm that produces a discrete joint distribution consistent with specifed values of the first four marginal moments and correlations is presented.

In the paper (Ponomareva et al., 2014) an algorithm for moment-matching scenario generation with application to financial portfolio optimization is given.

You may want to check also in http://www.itia.ntua.gr/en/docinfo/1656/ for an explicit (i.e. not through non-linear transformations but through simultaneous preservation of the distribution moments and the target correlation) algorithm of an exact simulation of a desired number of marginal moments (four or eventually even more) and any type of second-order dependence structure (via the second-order climacogram -i.e. variance vs. scale- or equivalently the correlation function).

• Thanks for this and welcome to CV. However, when posting links, it's good to also post information about the link, because sometimes links rot. – Peter Flom May 11 '18 at 11:36
• Hi! I have somehow edited my reply to include some informatiom on the merhod. I think both the papers you mention use non-linear transformation to generate the first four moments. Although the non-linear approach is excellent for short memory process I think is has some difficulties when it comes to long-memory processes (see for example a review on several methods in hal.archives-ouvertes.fr/hal-01399446/document) – pandim May 11 '18 at 17:05