Heavy Tail Index Estimation Based On Block Order Statistics

This paper mainly considers heavy tail index estimation when only a few order statistics are observed within blocks.Let {Xn,n?1} be a sequence of independent and identically distributed random variables.We divide the sample into several blocks,and construct estimators by using the order statistics in each block.The consistency and asymptotic normality of proposed new estimators in this paper are derived.Small sample simulations are presented to support our findings by comparison with two known heavy tail estimators established by block method.This papaer is mainly composed of three parts.For the first part,we construct a new kind of heavy tail index estimator by combining the traditional POT method with the BM method.This exploration is based on the work of Qi(2010)[41].The weak consistency of the estimator is derived.The asymptotic expansion and asymptotic normality of the estimator are considered under second order regular variation conditions.For the second part,we extend the quadratic Qi's estimator into a class of generalized heav,y tail index estimator,and also prove its consistency and asymptotic normality under second order regular variation conditions.Futhermore,the optimal choice of kn by means of minimizing Mean Square Error is discussed.For the third part,simulation studies are presented.Choosing suitable turning parameter ?,we compare the two type estimators proposed in this paper with the estimator proposed by Qi(2010)[41]and DPR estimator in terms of average mean,average mean squared error,coverage probability?and confidence interval length.Small sample simulation indicates that both proposed heavy tail index estimators show smaller estimated bias and mean square error,which supports the fact that the new estimators have good finite sample behaviours.