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The generalized gamma distribution includes the exponential distribution, 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. Prentice (1974) 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
finite summations.
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.

citation
Input File: gamma.omh