Asymptotic Properties Of Powered-extremes From Logarithmic General Error Distribution

This thesis mainly investigates asymptotic behaviors of powered-extremes of log-arithmic general error distribution sequences.Let {Xn,n? 1} denote a sequence of independent and identically distributed random variables with logarithmic general error distribution Fv(x),where v is the shape parameter.With Mn=max1?k?n(Xk)as their the partial maximum and Mnp as the powered-extremes where power index p>0.We have the following findings:When v? 1 the distribution of converges to the Gumbel extreme value distribution ?(x)under optimal choice of normalizing constants,and if v? 1 then it converges to the Frechet extreme value distribution ?(?)(x).The higher-order expansions and convergence rates of powered-extremes have no relation to the power index p.The latter shows that the uniform convergence rate of the distribu-tion of Mnp as p? 1 under different normalizing constants.The thesis is organized as the following two parts.In the first part,we mainly focus on the asymptotics of powered-extremes of loga-rithmic general error distribution as v>1.With power index p>0,the higher-order expansions of distribution of powered-extremes are established under the optimal nor-malizing constants,which shows that the distributions of powered extremes converges to the Frechet extreme value distribution at the rate of 1/n as v=1,and if v>1 then it converges to the Gumbel extreme value distribution at the rate of(log log n)2/(log n)1-1/v.The results show that power index p can't influence the convergence rate.In the secondly part,the uniform convergence rate of Mnp will be derived as p=1 and v>1 under different optimal normalizing constants,that is,the uniform con-vergence rate of distribution of normalized maximum is considered.Firstly,We derive asymptotic expansion properties of logarithmic general error distribution,then we find that the uniform convergence rate is of the order of 1/(log n)1-1/v.The results show that the convergence rates are different from the uniform convergence rates under different optimal normalizing constants.