Bahadur Representation And Asymptotic Normality Of The Quantile Estimator Of VaR Under α-Mixing Sample

After the nineties,in finance areas,VaR is a new important measure of market risk which has been accepted widely and has been applied importantly. Some large foreign financial institutions has applied its own value at risk VaR to regularly pub-lished financial statement. Therefore,VaR model has been the main model of mea-surement of financial market risk.So scholars do much research on VaR, study the estimator and properties of VaR.This note will discuss Bahadur representation and asymptotic normality of the quantile estimator of VaR,Yoshihara(1995,[2]) which derived the Bahadur representation by assuming that the strong mixing coefficantα(n)=O(n-β) whereβ>5/2,and the random variables are boundea, and the convergence rate is O(n-3/4 logn).Sun(2006,[24])tried to prove that the conver-gence rate of Bahadur representation of the quantile estimator is O(n-3/4+δlogn) underβ>10, whereδ∈(11/4(β+1),1/4),by removing the bound restric-tion.However,Sun(2006,[24])is not good as Yoshihara(1995,[2])and the order of strong mixing coefficient is required to beβ>10 instead ofβ>5/2,and Sun(2006,[24]) couldn't improve the convergence rate of Bahadur representation in Yoshihara(1995,[2]).Therefore,this note will discuss the Bahadur representation and asymptotic normality of the quantile estimator of VaR further,and got the better results than that in Sun(2006,[24]).Let{Y4}tn=1 be the market value of an asset over n periods of a time unit, and let Xt=log(Yt/Yt-1) be the log returns.Suppose{Xt}tn=1 is a strictly stationary depend process.Its marginal distribution function and density function are F(x) and f(x) respectively.Given a positive value p∈(0,1),the (1-p) confidence level VaR is We define estimation of VaR sample quantile estimation as and X(r)is the rth order statistic of X1,X2,…,Xn.The empirical distribution func-tion of X is whereⅠ(·) is an indictor function.We denotre qp=-vp,and Zn,p=-Qn·qp and Zn,p is the p-quantile of over and sample.Based on the definition above,in Chapter 2 and Chapter 3,we proved the follow-ing theorems. theorem2.1 Let{Xi,i≥1} be a strictly stationary sequence ofα-mixing random variables with a common distribution function F(x),where F(x)is ab-solutely continuous in some neighborhood of its p-quantie qp and has a continuous density function f(x)such that 0<f(qp)<∞.Suppose that 0<τ≤1 andα(n)=O(n-β),β>4/τ-3,Then,as n→∞,for anyτ>0 Zn,p-qp=o(n-1/2logτn),a.s.∞,for anyτ→0 where Jn={x:|x-qp|≤n-1/2logτn).theorem2.3 Suppose the conditions in Theorem 2.1 are satisfied and f'(x)is bounded in some neighborhood of qp. Then,as n→∞for anyτ>0In theorem2.3,whenτ=1,we derived the same result in Yoshihara(1995,[2]), where we only needβ>1 weaker thanβ>5/2 in Yoshihara(1995,[2]),and removed the bound restriction. Obviously,theorem2.3 improved the result in Sun(2006,[24]) greatly.In addition,we can prove the asymptotic normality as fol-lowing by using theorem2.3.theorem2.4 Suppose the conditions in Theorem 2.1 are satisfied.Ifβ>2,we haveAccording to the result in the theorem,the confidence interval of VaR estimation under confidence level 1-αis where u1-α/2 is the corresponding site in the normal distribution table.In Chapter 4,we choose two time series models which can produceα-mixing sequences,to test the accuracy of estimator VaR by numerical simulation under theα-mixing condition.In Chapter 5,according to the theoretical results of the second chapter,we do some analysis.