Minimum Variance Unbiased Maximum Likelihood Estimation of the Extreme Value Index

New results for ratios of extremes from distributions with a regularly varying tail are presented. Deriving from independence results for certain functions of order statistics, 'consecutive' ratios of extremes are shown to be independent as well as non-distribution specific. They have tractable distributions related to beta distributions. The minimum variance unbiased maximum likelihood estimator has the form of Hill's estimator. It achieves the Cramer-Rao minimum variance bound and is a function of a sufficient statistic. For small sample sizes the form of the moment generating function of the estimator shows it has a gamma distribution.