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I have a discrete distribution on $[0,\infty)$ that I can write down the algebraic expression for, but I can't compute it numerically for my desired parameters - basically, even a few probabilities would require too much computer time and memory, so it's computationally infeasible. It's really unfortunate because being able to compute the full distribution would be very helpful.

But I figured out a trick so that I can numerically compute as many moments of the distribution as I want! Is there a way to estimate a distribution based on its moments? What else can you do with only moments? Thank you!

bianca
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  • Would you be able to share this algebraic expression with us? – whuber Jan 11 '17 at 22:16
  • Sure, the pdf is $f(x) = \frac{1}{c} A^{x}_{i,j}$ where $A$ is a matrix, $c$ is a normalizing constant, and $i,j$ are parameters. In my application, the matrix $A$ is HUGE and that's what makes it computationally infeasible. – bianca Jan 11 '17 at 22:36
  • A matrix does not give one a probability distribution: how is $A$ related to the distribution you describe in the question? And if the problem is that it's huge, why isn't that also a problem for computing the moments? – whuber Jan 11 '17 at 22:41
  • As long as $\sum_{x=1}^{\infty} A^x_{i,j}$ is finite and positive, any matrix $A$ can give one a probability distribution like this (but please do correct me if I'm wrong). To be clear, by $A_{i,j}$ I mean entry $(i,j)$ of the matrix $A$. For some specific matrices, the expressions for the moments can simplify considerably, making it still difficult but more feasible than calculating the full distribution. – bianca Jan 11 '17 at 22:50
  • You lost me: although $A$ might be huge, $A_{i,j}$ is just a single number, which would seem unproblematic. Are you perhaps suggesting that $A_{i,j}$ cannot be calculated without also calculating the entirety of $A$? BTW, it is not the case that any matrix gives rise to a distribution in this sense. First, it depends on what values of $x$ are under consideration. Second, $A_{i,j}$ must be non-negative. I suspect you assumed that was understood. – whuber Jan 11 '17 at 22:53
  • Sorry, you're absolutely right, I definitely forgot to state things like the distribution support, positivity etc when I made that claim. And maybe I have failed to communicate something else: In $A^{x}_{i,j}$, the power is taken before the entry. So it is the exponentiation of a large matrix that is the core issue here. – bianca Jan 11 '17 at 22:58
  • Thank you--I did not understand that the superscript $x$ referred to an (integral) power. Your problem begins to sound like a Markov Chain calculation. Often the computation of a particular entry in a power of the transition matrix can be hugely simplified (through a preliminary decomposition of the matrix, if that's possible). So one way of addressing your problem might be to focus on an analysis of $A$ itself, rather than using the moments. If those moments are not known exactly, to extremely high precision, they might not do you much good. – whuber Jan 11 '17 at 23:01
  • It's a good point. I've looked into this using LU decomp and I think it won't work, because in order to calculate the distribution (and the constant $c$) to decent precision, it can require thousands of probability calculations. Your last sentence suggests that exact, high precision moments might be helpful - I'm not sure if mine will be precise enough, but out of curiosity, what could be done? – bianca Jan 11 '17 at 23:11
  • Distributions can be described by their characteristic functions and cumulant-generating functions. Both are power series whose terms are given by various moments. Provided you have enough moments and are not trying to peer too far out into the support of the distribution itself, you can hope to recover the individual probabilities from the cf or cgf. This is done by numerically computing a Fourier Transform. However, large moments are exquisitely sensitive to the larger values in the distribution's support, so great care is needed in using such approaches. – whuber Jan 11 '17 at 23:15
  • See https://stats.stackexchange.com/questions/141652/constructing-a-continuous-distribution-to-match-m-moments?rq=1 – kjetil b halvorsen Mar 02 '24 at 20:48

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