I tried to derive an expression for the log-gamma distribution using
the same procedure that can be used to derive an expression for the
log-normal distribution.
1. I started with the expression for a gamma [normal] distribution,
Gamma(x; alpha, beta) [Normal(x; mu, sigma)].
2. I replaced x with log y, yielding Gamma(log y; alpha, beta)
[Normal(log y; mu, sigma)].
3. I wrote out the expression for the integral of Gamma(log y;
alpha, beta) [Normal(log y; mu, sigma)] over (d[log y] = dy/y).
4. The integrand of the resulting integral over y should be the
log-gamma [log-normal] distribution.
This algorithm yields the right answer for the log-normal
distribution, but my result for the log-gamma distribution is different
from any form of the log-gamma that I have ever seen. Given a gamma
distribution of the form...
Gamma(x; a, b) = x^{a-1} * exp{-x/b} / {b^a * G(a)}...
...I obtain a log-gamma distribution of...
LogGamma(x; a, b) = [log x]^{a-1} * x^{-(b+1)/b} / {b^a * G(a)},
defined for 1 <= x <= infinity.
The possibilities are:
1. The relationship between the gamma and the (commonly used)
log-gamma distributions is defined differently than the relationship
between the normal and the log-normal distributions,
2. There is a common convention where the x in LogGamma(x; a, b)
stands for the exponent alone, a convention not commonly used with
log-normal distributions, or
3. I made a silly mathematical mistake someplace.
Does anyone know which possibility is the right one?
-- Andrew
.
.
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