Distribution sensitive estimators of the index of regular variation based on ratios of order statistics

Abstract: Ratios of central order statistics seem to be very useful for estimating the tail of the distributions and therefore, quantiles outside the range of the data. In 1995 Isabel Fraga Alves investigated the rate of convergence of three semi-parametric estimators of the parameter of the tail index in case when the cumulative distribution function of the observed random variable belongs to the max-domain of attraction of a fixed Generalized Extreme Value Distribution. They are based on ratios of specific linear transformations of two extreme order statistics. In 2019 we considered Pareto case and found two very simple and unbiased estimators of the index of regular variation. Then, using the central order statistics we showed that these estimators have many good properties. Then, we observed that although the assumptions are different, one of them is equivalent to one of Alves's estimators. Using central order statistics we proved unbiasedness, asymptotic consistency, asymptotic normality and asymptotic efficiency. Here we use again central order statistics and a parametric approach and obtain distribution sensitive estimators of the index of regular variation in some particular cases. Then, we find conditions which guarantee that these estimators are unbiased, consistent and asymptotically normal. The results are depicted via simulation study.