Are some representative moments of a probability distribution more representative than others?

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Alexander Andreychenko

I am supporting the answer from Theo K. Dijkstra, the distribution is not uniquely determined by the set of its moments. Some more information can be found in (and the corresponding references)

"Moment information for probability distributions, without solving the

moment problem, II: Main-mass, tails and shape approximation" by P.N. Gavriliadis and G.A. Athanassoulis, Journal of Computational and Applied Mathematics, 2009.

The context is needed to answer the question in more details (for instance, it might go further to the group of moment matching methods if you do not have any experimental data).

http://www.sciencedirect.com/science/article/pii/S037704270800513X

Diaa Al Mohamad

I was working on an optimization problem under constraints. The optimization was some divergence between a model and an unknown probability measure. The constraints should describe the unknown probability measure, but need to be applied on the model to ensure similarities. The unknown measure is a mixture. This is the reason why I wanted to describe the mixture distribution by some useful moments.

I don't ask for any kind of mixtures. I can add hypotheses to ensure that distribution can be represented by its moments.

On the other hand, since I'm working on an optimization under constraints, I need to reduce the number of constraints (number of moments used). This was the goal behinde my question. I thought that maybe using trigonometric moments, I might get more information by using a small number of constraints.

Thanks alot for your answers and the references therein.

Recently I have got to know about the maximum entropy method that allows to recover the density function from the given (finite) set of moments. There are many papers (mainly in the field of physics) where the authors are using Chebyshev moments and also exponential moments. However it seems to be not really related to your problem.

http://www.sciencedirect.com/science/article/pii/S0021999109004963

http://www.sciencedirect.com/science/article/pii/S0304407609002942

http://www.computer.org/csdl/proceedings/icmla/2008/3495/00/3495a421-abs.html