The Convergence Rate Of Extreme For Mixed Generalized Gamma Distributions

For a mixed generalized gamma random sequence,under an optimal choice of normalizing constants,the asymptotic properties of distribution and the uniform convergence rate of the normalized partial maxima are derived,the asymptotic be-haviour of observations near the maxima of such observations are also studied.This paper is mainly divided into two parts.In the first part,we discuss that the extreme for the mixed generalized gamma distribution converges to the domain of Gumbel for the parameters ci>1,λi>0,βi>0 and its uniform convergence rate.Then we discuss the asymptotic property of the neighborhood maximum Kn for the parameters 1>ci>0,λi>0,βi>0,Kn(a)weakly converges to a geometric RV taking values 1,2,...when ck = 1;kn(a)(?)∞,when Ck>1.And when 0<Ck<1,the mixed generalized gamma distribution is a thick tail;When Ck = 1,the mixed generalized gamma distribution is the middle tail;The mixed generalized gamma distribution is a thin tail when Ck>1.Finally,we get if x>(cκβκ-1/cκλkkc)1/cκ,Kn(a)(?)1 when 0<Cκ<1/2 and when 1/2≤Ck<1,Kn(a)(?)1In the second part,Due to the Mill rate limitation,the method of the uniform convergence rate in Hall et al.[18]is not suitable for calculating the uniform con-vergence rate of the mixed generalized gamma distribution with parameters λi = 1,ci = 1(written as G(1,β,1)).In this section,we cite another method to discuss the mixed gamma distribution(which is the mixed generalized gamma distribution with the parameters ci = 1,λi= 1,βi>0)converges to the domain of Gumbel,and its uniform convergence rate.