Fitting gamma distribution by moments and maximum likelihood ------------------------------------------------------------
^gamma4^ varname [^if^ exp] [^in^ range] [, ^s(^#^)^ ^t^ol^(^#^) l^og ]
Description -----------
^gamma4^ works on a single variable. All non-missing values must be positive. A two-parameter gamma distribution is fitted by the method of moments (a closed-form calculation) and by the method of maximum likelihood (ML) (an iterative calculation, which may occasionally be rather slow).
The algorithm used for ML estimation of alpha was proposed by P.W. Mielke. See Mielke and Johnson (1974) and Mielke (1976). The first approximation is that suggested by Thom (1958).
It is assumed that the parameterisation is 1. shape parameter, here called alpha 2. scale parameter, same units as data, here called beta.
Some people work with a scale parameter which in these terms is 1/beta. Yet others work with alpha and the mean, here called mu. Such people are recommended to treat the saved values suitably.
Note: this is the original version of ^gamma^, written for Stata 4. Users of Stata 8 up should switch to ^gammafit^.
Options -------
^log^ specifies that the results of each iteration should be displayed.
^s(^#^)^ controls the number of terms used within ^gamma4^ in a series approximation to the digamma function. The default is 100. This is a technical option and should not normally be changed. See Mielke (1976) for enlightenment.
^tol(^#^)^ controls the tolerance used within ^gamma4^ to control iteration. The default is 0.0000001. This is a technical option and should not normally be changed. See Mielke (1976) for enlightenment.
Example -------
. ^gamma4 precip^
Saved values ------------
S_1 number of values used S_alpha ML estimate of alpha S_beta ML estimate of beta S_amom moments estimate of alpha S_bmom moments estimate of beta
References ----------
Mielke, P.W. 1976. Simple iterative procedures for two-parameter gamma distribution maximum likelihood estimates. Journal of Applied Meteorology 15, 181-3.
Mielke, P.W. & Johnson, E.S. 1974. Some generalized beta distributions of the second kind having desirable application features in hydrology and meteorology. Water Resources Research 10, 223-6. See also 1976. Correction. Water Resources Research 12, 827.
Thom, H.C.S. 1958. A note on the gamma distribution. Monthly Weather Review 86, 117-22.
Author ------ Nicholas J. Cox, University of Durham, U.K. n.j.cox@@durham.ac.uk