Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

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Abstract
Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) bp–1 exp {–[(x – c)/a] p }/a bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results of applying this procedure to samples from Gamma, Weibull, and half-normal populations are tabulated, as are the asymptotic variances and covariances of the maximum-likelihood estimators.

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