Linear Estimation of the Scale Parameter of the First Asymptotic Distribution of Extreme Values

A Lagrange multiplier technique is used to obtain linear, minimum-variance, unbiased estimators for the scale parameters of the first asymptotic distributions of smallest and largest values with known mode. Coefficients for multiplying ordered observations are computed for complete and censored samples of size n = 1(1) 15. Each sample of size n is censored from above and all m-order-statistic estimators are obtained where m ¿ n. Then the smallest subset of # order statistics from the set of m available order statistics is found which yields a 99% efficiency relative to the m-order-statistic estimator. The Cramér-Rao lower bound for the variances of the estimators for complete samples is derived and tabled for n = 1(1) 15. For censored samples the asymptotic variances of the maximum-likelihood m-order-statistic estimators are presented for comparative purposes.