Extended Pattern Search with Transformations for the Three-Parameter Weibull MLE Problem

The Weibull distribution has found extensive applications in a variety of engineering, science and business problems. A journal survey indicated that it is the third most often used distribution in the statistical literature. To use it, one must best estimate its parameters from a given sample. Maximum likelihood estimates (MLE) of the three Weibull parameters are usually desirable due to their nice asymptotic properties, but several computational difficulties arise during the solution of this bounded-variable nonlinear optimization problem. This paper extends a method (pattern search) which was successfully used by the author for obtaining joint MLE of all three Weibull parameters. Fifteen different transformations (to eliminate bounds) and three search direction approaches (derivative-free, exact and numerical gradient) of a revised unconstrained pattern search are evaluated on a set of randomly generated realistic test problems. Performance criteria include computation speed, accuracy of estimators (relatively to the true parameters and true MLEs), log-likelihood value achieved, and goodness of fit of the estimated distribution (to the sample or generating population). The results are analyzed statistically and successful (unsuccessful) transformation/direction options are identified for each criterion. User interest in different performance criteria and the flatness of the log-likelihood function prevent identification of a single "overall best" option. Simple percentile estimators, used as search starting points, proved to be superior to MLE in problems with a small shape parameter value. The search procedures examined in this paper may also be used for solving other types of nonlinear optimization problems with bounded variables. They are attractive to practitioners because they are easily programmed and understood; they run fast but they do not guarantee optimality.