Distribution Sensitive Estimators Of The Index Of Regular Variation Based On Ratios Of Order Statistics

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