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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, how?

In particular, the main issue is, while it sees to be possible to convert from L-moments to the quantile function using Bernstein polynomials as a basis, the quantile function need not have the same dimension as the distribution.

The distribution, for example, could be defined on a sample space of integers from $1$ to $N$. Then, we would only need N raw moments to recompute the probability distribution, and likewise the cumulative distribution. However, the quantile function is the inverse of the cumulative distribution. While every point of the quantile function will hence be quantized from $1$ to $N$ - meaning the "range" is now what's sample - the domain need not be.

Does there exist a way in general to do this?

Mike Battaglia
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    Heuristically, the fact that this can be solved with $N$ moments is obvious: your random variable can be parametrized by $N$ unknown probabilities of each point, and you're solving a system of $N$ equations for $N$ moments. The only non-trivial part is showing the resulting matrix (A Vandermonde in this case) is non-singular. So heuristically you should be able to solve for $X$ in the $L$-moment case as well. The only thing to prove is that the resulting system is non-singular. – Alex R. Jan 12 '18 at 01:20
  • I'm not seeing how you get a Vandermonde matrix in the case of L-moments. It would be nice if it worked out like that, but I don't see how. The problem is that L-moments (and order statistics in general) are most directly related to the quantile function, which does have a nice decomposition into L-moments that can be described in terms of Legendre polynomials. Thus, you can get some kind of matrix decomposition for the quantile function. (continued) – Mike Battaglia Jan 12 '18 at 01:43
  • Now, it would be nice if we could get a matrix that goes from quantile function -> cumulative distribution -> probability distribution. And for the last step (discrete CDF -> discrete PDF), we can easily do this with a Toeplitz matrix. But going from quantile function to CDF is nontrivial, and I have no idea how you can use a matrix to do this. The spaces don't even have the same dimension. The domain of the CDF is the support of the probability distribution, but the domain of the quantile function is the *range* of the CDF. How would it work? – Mike Battaglia Jan 12 '18 at 01:48

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