Robust Parameter Estimation for the Mixed Weibull (Seven Parameter) Including theMethod of Minimum Likelihood and the Method of Minimum Distance

摘要

Robust parameter estimation is successfully applied to the Mixed Weibull (sevenparameter) using the Method of Minimum Distance and the Method of Maximum Likelihood. That is, parameters can now be estimated for a mixture of two Weibull distributions where the true populations are co-located, partially co-located or highly separated. Both techniques provided very robust estimates that were far superior to current parameter estimation techniques. Sample sizes as low as ten with mixing proportions down to 0.1 were investigated. For the MLEs, innovative bounding techniques are presented to allow consistent and correct convergence using any reasonable point estimate. The likelihood function is solved numerically as a non-linear constrained optimization using a quasi-Newton method. Minimum Distance Estimates (over three hundred scenarios investigated) are derived for some variation or combination of the mixing proportion and the location parameter(s), individually and simultaneously (the Anderson-Darling and Cramer-von Mises statistics were used). In tact, the MDE for the mixing proportion was so effective that future researchers should consider some permanent combination Primary measures of success were based on comparison of CDFs. Mean square error (MSE) and integrated absolute difference (LAF) between the estimated and true distributions were measured including confidence intervals.