Generalized gamma parameter estimation and moment evaluation
The generalized gamma distribution includes the exponential
the gamma distribution, and the Weibull distribution as special cases.
It also includes the log-normal distribution in the limit as
one of its parameters goes to infinity.
developed an estimation method that is effective even
when the underlying distribution is nearly log-normal.
He reparameterized the density function so that it
achieved the limiting case
in a smooth fashion relative to the new parameters.
He also gave formulas for the second partial derivatives
of the log-density function to be used in the nearly log-normal case.
His formulas included infinite summations, and he did not
estimate the error in approximating these summations.
We derive approximations for the log-density function
and moments of the generalized gamma distribution that
are smooth in the nearly log-normal case and involve only
Absolute error bounds for these approximations are included.
The approximation for the first moment is applied to the
problem of estimating the parameters of a generalized gamma distribution
under the constraint that the distribution have mean one.
This enables the development of a correspondence between
the parameters in a mean one
generalized gamma distribution and certain parameters
in acoustic scattering theory.