My problem is to generate a univariate discrete distribution
having specified first four moments. This is part of a larger
problem where I want to generate a multivariate discrete
distribution where the moments are specified separately for
each dimension and the multivariate distribution should also
conform to a specified correlation matrix. I have an algorithm
(a heuristic) that is able to split the large task into the task
of generating the univariate distributions separately and then
modifying them to reflect the desired correlation matrix, while
preserving the specified moments to good accuracy. This heuristic
requires the use of the same discrete probability mass
distribution for all the variates, so the small problem reduces
to choosing the outcomes while the probability masses are
specified beforehand. The billion dollar question is now how
to choose the probability masses and how to specify the objective
function to minimize. I currently use 5 outcomes (this is enough
for 4 specified moments) and minimize the sum of squared differences
of specified moments and discrete distribution moments. I get
better results when specifying the probability masses according
to an approximate normal distribution than choosing them all to
be the same (my data that the moments are extracted from are
leptocurtic by the way).
Another difficulty that has appeared is that with only 5 outcomes
the implied correlation matrix is not close to the identity
matrix (I want uncorrelated univariate distributions),
typical correaltions are of the size .5
By choosing a larger number of outcomes the generated univariate
distributions are less correlated, but there is still
correlation. How should I do to reduce the correlation ?
Any hints, comments or pointers to relevant research literature
is very welcome.
Tomas
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