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.