FURTHER RESULTS ON ESTIMATION OF THE PARAMETERS OF THE PEARSON TYPE III DISTRIBUTION IN THE REGULAR AND NONREGULAR CASES.
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The report is devoted to a continuation of research on problems in nonregular estimation. In previous reports the major emphasis was on estimation of the location parameter of life-distributions, such as the Pearson Type 3 and Weibull distributions, for which certain regularity conditions required for optimality of the maximum likelihood estimators are not satisfied over the entire parameter space. The regularity conditions considered were those of Cramer. For the Type 3 distribution, the nonregular case was defined to be that in which the shape parameter, alpha, is less than or equal to 2. In fact, the Cramer conditions are not satisfied for any alpha. Investigation of alternative conditions in the case of Type 3 and Weibull distributions is pursued in this report. It is found that these are also not satisfied, but it is conjectured that somewhat weaker conditions, which are satisfied if the shape parameter of either distribution exceeds 2, will suffice. A second aspect of the estimation problem for the Type 3 distribution discussed in this report is the estimation of all three parameters location, scale and shape. Finally, this report includes further results on application of a nontrivial lower bound on the variance of unbiased estimators to the Type 3 distribution.
- Statistics and Probability