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Topic: Gamma parameters
Replies: 0

 Gaines Harry T Posts: 22 Registered: 12/12/04
Gamma parameters
Posted: Jul 1, 1996 4:32 PM

I am interested in estimating the parameters, alpha and beta (using the
notation of Korn & Korn) of a gamma distribution and placing confidence
limits on them, or testing hypotheses about them. I couldn't find just
what I was looking for, but it wasn't hard to develop maximum likelihood
estimates. Denoting the parameters by a and b, I found that the MLE of
the product ab is the arithmetic mean of the data, a natural result since
this product is the mean of the population. I also found the equation for
the MLE of a. It states that the ratio of the arithmetic mean to the
geometric mean equals the following function of a: f(a) = a*exp(-psi(a)).
Here. psi denotes the logarithmic derivative of the gamma function. One
can solve the equation numerically to find the MLE of a and then of b.

I also learned (from a colleague and confirmed in Feller) that the sum of
independent gamma variables with the same b is a gamma variable with the
same b and with a the sum of the a's. This lets me get a handle on the
distribution of the arithmetic mean if I am willing to some make
assumption about a or find the distribution of the MLE of a.

I'm not a statistician, and I'm sure I've been reinventing the wheel. I
don't see any easy answer, so I'm asking for help. Can someone tell me,
or give me an accessible reference that can tell me about the distribution
of MLE of a and b? Are they independent or must I deal with their joint
distribution?

I would greatly appreciate e-mail responses (in addition to posted
replies) because I'm afraid responses might expire before I find them.