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st: An application of the extended generalized gamma function

 From Francisco Augusto To statalist@hsphsun2.harvard.edu Subject st: An application of the extended generalized gamma function Date Thu, 30 Aug 2012 18:01:50 +0100

```Dear Statalist,

I have sent this question before, but since it laked a considerable
ammount of expressions and reference, I am sending the question again
hoping for a possible anwser.

This is the problem: I have a variable x, which is in logarithmic
scale, that I want to adjust to the extended generalized gamma
distribution following the approach used by

R.L. Prentice 1974, "A log gamma model and its maximum likelihood
Estimation", Biometrika, 61(3), pp 539-544

and developed by

J.F. Lawless 1980, "Inference in the generalized gamma and log gamma
distributions" Technometrics, 22(3) pp 409-419.

in such a way I may replicate the work developed by

Cabral & Mata 2003, "On the Evolution of the firm size
distribution:Facts and Theory", The American Economic Review, 93(4),
pp1075-1090.

The first approach I considered was the streg command, which is not
suitable for this problem, since there is a little change in the
original expression of the extended generalized gamma function:
starting from the original expression and considering the three
parameters (mu, sigma, k). The parametrezation that I present
considers the following transformation q = k^(-1/2), leading to the
following expression (citing from Cabral and Mata 2003):

if x follows an extended generalized gamma distribution, then w = (lnx
- mu) / sigma has p.d.f.

|q|*(q^-2)^(q^-2) *exp(q^(-2) *(qx - exp(qx))) / gamma(q^(-2)) if q is
different from 0

(2pi)^(-1/2) èxp(-(1/2)x^(2)) if q is equal to 0

where gamma(t) is the gamma function.

My objective is to estimate the three parameters (mu, sigma, q) and
then to regress on x assuming that x follows an extended generalized
gamma function (exactly the same as Cabral and Mata 2003 did)

(Sorry for the messy expression...). I am using Stata 11.0.

I am open to any suggestion and please correct me in case of mistakes.
If any other information is needed, I will be please to add it to the
question.