# [SciPy-User] Kurtosis/Skewness

nicky van foreest vanforeest@gmail....
Tue Mar 30 16:32:24 CDT 2010

> There are an infinite number of distributions that will have the same
> skewness and kurtosis. However, it is reasonable to search for the
> maximum entropy distribution satisfying those constraints. The normal
> distribution is the maximum entropy distribution for a fixed mean and
> variance.
>
> http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution
>
> The PDF will have the form:
>
>  pdf(x) = c * exp(- lagrange * (x ** arange(1, 5)))
>
> c is just the normalizing constant. You will have to find the lagrange
> parameters that satisfy the mean, variance, skewness and kurtosis.
> Sampling from this distribution will be tricky, though. You will have
> to resort to general methods that are going to be pretty slow.

This is of course a very good suggestion. However, mind that this
claim is only true if the support of your desired distribution is the
entire real axis. I recall that I once tried to find the maximum
entropy distribution with given mean and variance, but such that the
support was the positive reals (including 0), rather then the entire
real line. This was less easy then I initially thought, and it is
certainly not the normal distribution.

bye

Nicky