The Kernel Estimation Of Expected Shortfall Under The Statistic Orders

VaR(Value at Risk) has being considered as a very important risk measure in the marketrisk management.In 1996, However, Artzner et.al first presented the concept of coherentrisk measures in 1997, and pointed out that VaR is not a coherent risk measures because itdoesn't satisfy the sub-additive. In order to construct a risk measure which is both coherentand easy to compute and to estimate,the Expected Shortfall (ES)was proposed and discussedby Acerbi et.al[11].So, ES has more study value as a coherent risk measure.In 2004, Scaillet[20][21] presented a two-step nonparametric estimation, Moreover ,there is much work in Chen [16], establish the Bahadur representation , get the Expectationand Variance , and the convergent rate of them respectively: O_p(n^-1) + o(h²),op(h²) +O_p(n^-1),o(n^-1h).In this paper , we develop a new nonparametric estimation of ES un-der theα-mixing series (the kernel estimation of expected shortfall under the statisticorders),establish the Bahadur representation, get the Expectation and Variance, otherwiseAsymptotic normality of Kernel-type quantize estimator is given. The convergent rate ofthem respectively:o(n^-3/2h^-2 log^3τ n),o(h²) + O((nh)^-1),o(n^-1h^1/4).We compare the kernelestimation of expected shortfall under the statistic orders to the two-step estimator their bi-ases is small, which shows that the kernel estimation of expected shortfall under the statisticorders has a good risk measure as the same.There is some work of ES and VaR last years. Xu xusong [5]empirically comparesES with VaR from aspects of convexity and validity and validity on condition of normaldistribution and non-normal stable distribution on China Stock Market. Chen shoudong[2]computing VaR and ES based on EVT, Ou shide[4]empirically compares ES with VaR byusing the last nonparametric estimation of VaR and ES, the result is good. In this paper ,wecompare the kernel estimation of expected shortfall under the statistic orders with two-stepnonparametric estimation by using nonparametric estimation, analyze empirically the risk ofShanghai Stocks index which from 2001.1.2 to 2006.12.29, their biases are small, we alsocompare the ES and VaR under the kernel estimation of expected shortfall under the statisticorders, which shows that the kernel estimation of ES is better than the kernel estimation ofVaR on China Stock Market. The paper is structured as follows:In chapter 1, some knowledge about VaR and research about kernel quantile estimatorare first introduced , and given the theoretical models and the nonparametric estimation ofVaR and ES.In chapter 2, the concept ofα?mixing and the four basic assumption are given, meanwhile, the main results and several lemmas which are needed in the paper areshowed.Some lemmas are the work of predecessors, some are proved in the paper basedon the past work.For example , in lemma2.2.2and lemma2.2.4.In lemma2.2.2,we prove theChen[16]by using Monotonicity and HolderInequality. The method gets rid of the limit thatChen[16]gives.Meanwhile,In lemma2.2.4,the proof is more accuracy because we have con-sidered the convergence rate.In chapter 3,the part is the main proof process. we derived the mean, the variance andthe mean square error of the kernel estimation of expected shortfall under the statistic orders,and prove the asymptotic normality.In the proof process, we considered the iterated logarithmthat aspects the convergence rate .Using lemma2.2.2and lemma2.2.4,we derive some of thekernel estimation of expected shortfall under the statistic orders .In chapter 4, VaR and ES. we compare the kernel estimation of expected shortfall underthe statistic orders with two-step nonparametric estimation by using nonparametric estima-tion in different quantiles. We also compare the VaR and ES under the kernel estimation ofexpected shortfall under the statistic orders ,which shows good results.In chapter 5,we summarize the main results of this paper and show that there are manyproblems we will study under the models of the kernel estimation of expected shortfall underthe statistic orders,for example, the strong consistence.Some features under other mixingSequences,and the using in actuary studies and so on.