Sufficient Statistical Estimation Functions for the Parameters of the Distribution of Maximum Values

Abstract

The problem of estimating from a sample a confidence region for the parameters of the distribution of maximum values is treated by setting up what are called "statistical estimation functions" suggested by the functional form of the probability distribution of the sample, and finding the moment generating function of the probability distribution of these estimation "functions. Such an estimate by the method of maximum likelihood is also treated. A definition of "sufficiency" is proposed for "statistical estimation functions" analogous to that which applies to "statistics". Also the concept of "stable statistical estimation functions" is introduced. By means of a numerical illustration, four methods are discussed for setting up an approximate confidence interval for the estimated value of $x$ of the universe of maximum values which corresponds to a given cumulative frequency .99, for confidence level .95. Two procedures for solving this problem are recommended as practicable.