Bahadur Representation And Asymptotic Normality Of The Estimator Under P-Mixing Sample

VaR(Value at Risk) is an important risk measure which emerges in the 90's. AlthoughVaR only satisfies the sub-additive of the coherent risk measure on the condition that there islittle loss probability, however, actually, people only care about a loss probability in practice.Besides, according to the massive practical application, we find it easer to calculate withVaR compared with other risk measure. Therefore, in recent years, people consider VaR asan ideal risk measure gradually, and launch to do the massive research on it. The samplequantile, as the nonparametric estimation of VaR forρ-mixing sample, this note not onlyexplains the asymptotic normality of Bahadur, but also does some numerical simulation andempirical study to obtain some valuable results.Let{Yt}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 dependent process.Its marginal distribution function and density function are F(x) and f(x) respectively. Givena positive value p∈(0, 1), the (1 - p) confidence level VaR isvp = -inf{u : F(u)≥p}. (0-5)We define the estimation of VaR sample quantile estimation asQn,p = -X([np]+1),where X(r) is the rth order statistic of X1,X2,···,Xn. The corresponding empirical distri-bution function iswhere I(·) is an indicator function. In the whole article, we denote qp = -vp, and Zn,p =-Qn,p. In fact, qp and Zn,p is the p-quantile of overall and sample.To get the theoretical result, we assume:Assumption1 {Xi : i≥1} is a strictly stationary sequence ofρ-mixing random variables,ρ(n) = O(n-β), whereβ> 1.Assumption2 F(x) is a common distribution function, and is absolutely continuous insome neighborhood of its p-quantile qp.Assumption3 f(x) is a continuous density function, 0 < f(qp) <∞,where p∈(0,1), and it has continuous first derivatives in B(qp) which is a neighborhood of qp . Fkis the joint distribution function of (X1,Xk+1)(k≥1), and it also has its second partialderivatives bounded in B(qp).In Chapter 2, we prove that the Bahadur representation and the asymptotic normality aswell as uniformly asymptotic normality of the VaR sample quantile estimation.THEOREM 1.1(Bahadur representation) Let A1-A3 hold,and f (x) bounded in aneighborhood of qp. as n→∞where 0 <τ≤1.Remark1: Yoshihara(1995, [1]) derived the Bahadur representation by assuming thestrong mixing coefficientα= O(n-β) whereβ> 5/2. The condition ofβ> 1 in A1 of thispaper is weaker thanβ> 5/2 of Yoshihara(1995, [1]).As n→∞,σp2 converges absolutely under the conditions of A1-A2. see Page x. Denoteσp2 = limn→∞σp2,n, and Un =√nf(qp)(Zn,p - qp)/σp.THEOREM 1.2(Asymptotic normality) Let A1-A3 hold. Ifβ> 2, we have√n(vp - Qn,p) -→d N(0,σp2f-2(qp)).Remark2: the confidence interval of VaR estimation under confidence level 1 -αis[Qn,p - u1-α/2 nfσ(pqp),Qn,p + u1-α/2 nfσ(pqp)],where u1-α/2 is the corresponding site in the normal distribution table.THEOREM 1.3( Uniformly asymptotic normality) Let A1-A3 hold. If 0 < b < 1 andthen for any > 0whereΦis the standardize normal distribution function and FX(s) denotes the distributionfunction of any random variable X. Remark3:β≥7/5 as b→1, and if≥1/4, we haveIf < 1/4,the convergence rate of uniformly asymptotic normality is near to n-1/4;β≥2as b→0, and if≥1/12, the convergence rate of uniformly asymptotic is near to n-1/6, if< 1/12, its convergence rate is also near to n-1/4.In Chapter 3 , this note does the numerical simulation of the VaR sample quantile esti-mation ,and tests its accuracy.In Chapter 4, according to the theoretical results of the second chapter, we analysis theShanghai index and Shenzhen index from January 4, 2006, to December 28, 2007. we usethe VaR sample quantile estimation to assess the risk of the two markets, and compare therisk and the VaR estimation's confidence interval of these markets during different periods.