Asymptotics On Powered-Extremes Of Generalized Error Distribution Sequences

This paper is devoted to investigate the asymptotic behaviors of powered-extremes of generalized error distribution sequences.Let {Xn,n?1} denote a series of independently and identically distributed random variables with generalized error distribution Gv(x),where v is the shape parameter.With Mn=max1?k?n(Xk)as their partial maximum and |Mn|p as their powered-extremes where power index p>0,the following findings are derived from this paper:The distribution function,density function and moment of|Mn|p converge to the corresponding distribution,density and moment of Gumbel extreme value distribution under optimal choice of normalizing constants.And the choice of power index p can influence the convergence rate.This paper is divided into three parts:In the first chapter,the higher-order expansions of distribution of powered-extremes of GED sequences will be discussed.The results show that under optimal normalizing constants,the convergence rate is proportional to 1/log n as p ?v,and the convergence rate can be improved to 1/(logn)2 diS p=v under optimal normalizing constants.In the second chapter,the higher order expansions of density function of powered-extremes of GED sequences will be derived as p?v and p=v under different optimal normalizing constants,respectively.The results show that the density function converges to the Gumbel type density function at a rate proportional to 1/(logn)2 when p=v under according optimal normalizing constants.In the third chapter,the higher order expansions of moment of powered-extremes of GED sequences will be derived as p t,and p=v with different normalizing constants.And the moment converges to the Gumbel type moment at a rate proportional to 1/(logn)2 when p=v under according optimal normalizing constants.