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Estimation of the upper limit of finite range distributions
This report is concerned with the problem of estimating the upper limit of a distribution using a random sample drawn from it. Specifically, the theory may be applied to the estimation of either the upper or lower limit of finite range distribution or to the estimation of either the upper or the lower limit for distributions where one of the limits is infinite. The literature on this particular estimation problem is limited and confined to special cases of parametric distributions such as the gamma distribution with unknown origin and the uniform distribution. The most general situation discussed in the literature arises in the Pearsonian system of frequency curves with the lower end or upper limit finite. The moment based theory as described in the classic "Frequency Curves and Correlation" by H.P. Elderton can of course be applied to the present problem and this forms indeed the first approach developed in this report. However Elderton's formulas only provide point estimates and it was therefore necessary to develop variance formulas for the estimates using the theory of kappa parameters and k-statistics. Because of the transcendental dependence of the estimators of the k-statistics approximate variance formulas based on statistical differentials would have to be developed. Because of certain unsatisfactory properties of these moment based estimators a new approach is developed for the important case of the Type I (beta) distribution having four parameters; namely the upper and the lower limits as well as the two exponents. Here the Elderton moment based estimators for the exponents were retained but used in conjunction with either the first or last order statistic to obtain estimates of the upper or lower limit. Again, the method of "statistical differentials" is used to obtain approximate variance formulas. A comparison of the two methods either by a comparison of the approximate variance formulas or by Monte Carlo simulation has established the superiority of the second method of estimation.
"Research conducted through the Texas A and M Research Foundation."
Sponsored by the Office of Naval Research N00014-68-A-0140 NR047-700
Includes bibliographical references (leaf 85).
Technical report (Texas A & M University. Themis Optimization Research Program) ; no. 40