    # st: RE: Creating a distribution from moments

 From "Nick Cox" To Subject st: RE: Creating a distribution from moments Date Fri, 3 Nov 2006 12:08:39 -0000

```This is called the method of moments
and Karl Pearson thought, more or less,
that it was the best way to fit a distribution,
so long as you know what kind of distribution
you are fitting. But it has never really recovered
from criticisms made by Ronald A. Fisher and
others several decades ago, except that we
still use it routinely in problems like plugging
the mean and the standard deviation into a
Gaussian (so-called normal) formula.
(That is usually very close to the maximum
likelihood solution, but most summary programs,
like -summarize-, use n - 1 not n as divisor
for the variance estimator.)

Yes, you can have a go at this, but you need to
look at books like the multivolume reference
by N.L. Johnson, S. Kotz and friends to see the ways
to do it. Usually there is some indirectness:
for example, I can't recall any named distribution
for which one of the parameters _is_ the
skewness or the kurtosis: rather if you have
k parameters, you typically end up with k
simultaneous equations in those parameters
and the first k moments will be useful in
solving those.

(Strictly, there is a terminology problem here:
the skewness and kurtosis, at least as
named in Stata, are ratios derived from
moments, not moments themselves, but we
know what you mean.)

More important, this is only the method of
choice if these summaries are _all_ you
have to go on. If you have the data, use
all the data.

Nick
n.j.cox@durham.ac.uk

Reza C Daniels

> I have summary statistics for four moments of the density of y:
> mean=877; std. deviation=611; skewness=0.658; kurtosis=2.278. Is it
> possible for me to use this information to generate a hypothetical
> density whose four moments approximate these values?
>
> The analogy here would be creating, for example, a Gaussian
> distribution
> by:
>
> set obs 1000
> gen z1=invnorm(uniform())
>
> My question relates to when one does not want a standard density but
> possibly some parameterization of a Beta or Gamma that fits the four
> moments.
>
> I have searched through the probability distributions and density
> functions in Stata (I'm using version 8.2SE), but it is not
> immediately
> obvious to me that I can do this.

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```