ON MAXIMUM LIKELIHOOD ESTIMATORS OF SHAPE AND SCALE PARAMETERS AND THEIR APPLICATION IN CONSTRUCTING CONFIDENCE CONTOURS

In a paper by Thoman, Baine and Antle, it was shown that certain pivotal functions of the maximum likelihood estimators for parameters in the Weibull family have distributions which are parameter-free. By using Monte Carlo methods, this basic result made possible the production of tables of the percentile points of these distributions so that confidence intervals for the parameters can be determined. In turn this makes possible tests of hypotheses regarding the parameters. It is the purpose of this note to generalize this result in several respects. Firstly, to point out that for any given continuous distribution with support on the positive line and unknown shape and scale parameters the same pivotal functions of the maximum likelihood estimators are parameter-free. Secondly, that the same conclusion holds for incomplete samples censored under certain conditions or terminated at a given ordered observation.