Parameter Estimation for a Generalized Gamma Distribution

It is fairly commonplace in reliability analyses to encounter data which is incompatible with the exponential, Weibull, and other familiar probability models. Such data motivates research to enlarge the group of probability distributions which are useful to the reliability analyst.

In this paper, we examine a three-parameter generalization of the gamma distribution and derive parameter estimation techniques for that distribution. Those techniques, in the general case, depend upon method of moments considerations which lead to simultaneous equations for which closed form solutions are not available. Graphic solution is proposed and aids to the computations are provided. Major concepts in the paper are summarized by means of a numerical example.

Details are given for the special case in which only the scale parameter is unknown. Three unbiased estimators for that parameter are derived along with their variance formulas. Minimum variance considerations are discussed by application of the Cramér-Rao Theorem.     

Complete Sample Estimation Techniques for Reparameterized Weibull Distributions

The Weibull distribution, frequently employed to assign probabilities to the lifetimes of components and systems operating under stress, is habitually characterized by a pair of positive parameters, termed the scale and shape parameters. Two fundamental reparameterizations of the Weibull probability density function are proposed. The first reparameterization replaces the shape parameter by its inverse, the resulting positive parameter thereafter termed the shaping parameter. This permits a more facile exposition of the properties of parameter estimates, derived in the event that a complete random sample from the Weibull distribution is available. The characteristics of these parameter estimation techniques are then reviewed and compared, and their variances and distributional properties are delineated whenever possible. A second reparameterization extends the parameter space so as to include nonpositive values of the shape parameter. This extension augments the utility and applicability of the Weibull distribution without requiring radical alteration of the standard parameter estimation procedures applicable to the original parameter space.