#StackBounty: #matrix #moments #method-of-moments #l-moments Discrete, finite L-moment problem

Bounty: 50

Suppose that we have a real-valued discrete random variable, whose probability distribution has finite support on some set \$S\$ of real numbers. Then if \$N = |S|\$, we know that we can construct the entire distribution from the first \$N\$ raw moments, as described in this paper:

https://www.sciencedirect.com/science/article/pii/0166218X9090068N

The transformation is a simple Vandermonde matrix that converts from moments to probabilities.

Suppose that we instead want to use the L-moments. Is there an analogous result where we can completely reconstruct the distribution using only the first \$N\$ L-moments, and if so, what does the resulting matrix look like?

To be specific, I am looking for the basis specified by the matrix that solves this problem in the discrete, finite case. For the classical (raw) moments, the basis is the monomials up to order \$N\$.

I know that Bernstein polynomials are often mentioned in connection with the L-moments, although I’m not sure if this helps here.

I also understand that we can use the L-moments to reconstruct the quantile function, but I’m not sure how many L-moments are needed to reconstruct the entire thing, nor how this translates into a basis for the discrete finite probability distribution.

Get this bounty!!!