Comments on: How well do moments determine a distribution?

By: John

John — Mon, 19 Nov 2012 23:24:47 +0000

For a particular set of moments, you can calculate the matrix M and get P(x) exactly. The paper I reference gives specific examples.

But in any case, you know that asymptotically the difference between the two distribution functions is O(1/x^2p). The reason moments tell you more in the tails than in the middle is that for sufficiently large values of x, only the leading term in the polynomial matters.

By: Mirek Kukla

Mirek Kukla — Mon, 19 Nov 2012 22:28:02 +0000

I’m not exactly sure how to interpret this result.

On the one hand, the title of the paper is “Moments determine the tail of a distribution (but not much else).”

On the other hand, the fact that there is a polynomial P(x) where |F(x) – G(x)| ≤ 1/P(x) doesn’t tell us much if we don’t know what P(x) looks like. Moreover, this is an upper bound, which limits the dissimilarity of the distributions.

In other words, the title of the paper implies that moments don’t determine a distribution very well, whereas the result allows you to conclude nothing more than that moments *might* match a distribution quite well.

What’s the takeaway, then?

By: Vladimir Bakhrushin

Vladimir Bakhrushin — Mon, 19 Nov 2012 16:07:40 +0000

This is an interesting result. But I would like to pay attention to one circumstance – it is assumed that moments which we know, we know exactly. In fact, as a rule, it is not so. And it would be interesting to recieve estimates, taking into account the errors of moments.

By: John

John — Sun, 18 Nov 2012 19:00:26 +0000

Jonathan: You multiply by V on the left and the right. That’s why it’s degree 2p. For example, if the matrix in the middle were the identity, you’d get 1 + x^2 + … + x^2p.

By: Jonathan

Jonathan — Sun, 18 Nov 2012 16:31:26 +0000

If V is of dimension p+1, wouldn’t this make P a polynomial of degree p?

By: John

John — Sat, 17 Nov 2012 23:30:08 +0000

David: Your comment made me realize I’d incorrectly written the dimensions of V and M. The matrix M depends only on the 2p moments in common between X and Y.

I updated the post. Thanks for pointing out the error.

By: David Feuer

David Feuer — Sat, 17 Nov 2012 22:54:04 +0000

If I read this summary correctly, if two distributions have the same first 2p-1 moments, then if you know the first 4p moments of one of the distributions, you can determine inverse-polynomial bounds for the difference between them. Those bounds tend to infinite width near the shared mean (revealing less than the trivial bound |F(x)-G(x)|≤1). The larger the higher moments of the chosen distribution, the faster the bound narrows. That's how I read it anyway.

By: Jan Van lent

Jan Van lent — Sat, 17 Nov 2012 14:13:03 +0000

I haven’t read the linked article, but I think the following Wikipedia links for the moment and truncated moment problem are relevant.
The second link shows a connection with orthogonal polynomials.

http://en.wikipedia.org/wiki/Moment_problem
http://en.wikipedia.org/wiki/Chebyshev%E2%80%93Markov%E2%80%93Stieltjes_inequalities

By: Jan Galkowski

Jan Galkowski — Sat, 17 Nov 2012 06:21:02 +0000

Yes, agree with Mr Noble: This is neat, but to be appreciated beyond the “priesthood”, needs to be wrapped in explanatory language.

By: Steven H. Noble

Steven H. Noble — Sat, 17 Nov 2012 03:58:32 +0000

The title of that paper really helps put this result in context.