ESTIMATION OF THE LOCATION PARAMETERS OF THE PEARSON TYPE III AND WEIBULL DISTRIBUTIONS IN THE NON-REGULAR CASE AND OTHER RESULTS IN NON-REGULAR ESTIMATION

Abstract

Abstract : The project is a continuation of research on problems in non-regular estimation reported in ARL Technical Documentary Report No. ARL 65-177(1965) AD- 622 274. Included in that report was a lower bound on the variance of unbiased estimators of the location parameter of the Pearson Type III distribution, applicable in the non-regular case. This report includes the results of a numerical investigation of that bound for varying values of the shape parameter of the Type III distribution and varying sample sizes. The bound is apparently of the correct order of magnitude in a certain region of the parameter space by sub-optimal elsewhere. Approximations to the Pitman estimators for location parameters are investigated for both the Pearson Type III and Weibull distributions. In both cases, the minimum observation apparently contains the major part of the information concerning the unknown location parameter. Some results on the non-regular estimation problem, particularly concerning the derivation of variance bounds, in the cases of densities with bounded domain depending on an unknown parameter and of mixtures of uniform distributions, are also discussed.