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st: RE: Creating a distribution from moments

From   "Nick Cox" <[email protected]>
To   <[email protected]>
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. 

[email protected] 

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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