A Class Of Heavy Tailed Index Estimator Of Location Invariant Based On Hill Estimate

Many empirical studies have shown that events in life that does not meet the normal distribution show diversity,they can be well expressed by heavy-tailed distributions.Such as mathematical statistics,geography,meteorology,insurance and risk theory,etc.,in order to understand the regularity behind the phenomenon,we should make an estimation of heavy tailed index,which has important practical value to risk aversion and forecast.The effectiveness and robustness of the tail index estimator have become more and more important in recent years.On the basis of extreme value theory and heavy-tailed distribution,This paper introduces several kinds of classical estimators and related properties,then lead to a location invariant hill-type estimator and related properties,at the same time,making use of asymptotic expansion of the statistic M_n^a?k₀,k?,the article puts forward a new location invariant extreme value index???_n¹?k₀,k,a?,which introduced a tuning parameter a,and in second order regular variations,deduces the asymptotic properties of the estimator above,and proposes a method of selection on the threshold k₀.At last,from the angle of asymptotic efficiency relative to control the tuning parameters a,it ensures we get an estimator with principal component deviation estimator of close to zero,then using Monte-Carlo method to make a simulation and comparative analysis tocompare the estimator???_n¹?k₀,k,a?and???_n¹?k₀,k,a? proposed by Fraga Alves,using three different kinds of distribution Fréchet,Pareto and Burr,from the angles of mean value and mean square error to evaluate the superiority of this two estimators.The results show that the new estimator behaves better.