Re: Estimating Gamma Parameters using MLE
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 23 Oct 2005 16:39:08 -0500
In article <1129912850.457338.131520@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<kipper_fillet@xxxxxxxxxxx> wrote:
>A collegue needs to put together some Visual Basic (ideally) code to
>perform maximum likelihood estimation of the parameters of the gamma
>distribution (based on a series of observations passed across as an
>array).
>Does anyone have such a thing, or know where it could be obtained?
This is ancient history. The density function is
x^(q-1)*exp(-x/s)/(s^q*Gamma(q),
and
\int x^(q-1)*exp(-x/s) dx = s^q*Gamma(q).
The sufficient statistics are the sums of the X's and
the sums of their logarithms, and the MLE sets the
sample averages to their expected values. The expected
value of X is q*s, and the expected value of ln(X) is
ln(s)+Psi(q), the Psi function being the logarithmic
derivative of the Gamma function. So one sets the
average of ln(X_i) minus the logarithm of the average
of the X_i to be equal to Psi(q) - ln(q); this has a
unique positive solution if the X's are not all equal,
and finding the MLE of s is then easy from the average
of the X's.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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