| In economic and finance areas, it is very important to avoid risks, so someone arise thathow to avoid risks and how to understand the degree of future risks, how to describe thedegree of future risks and what can measure the risks in the future and so on. Based onthese thoughts, scholars found that such as standard deviation, absolute deviation, Value atRisk(VaR), ES, CVaR and so on all can describe the degree of future risks from di?erentperspective. A lot of work found that each estimator has its advantages and disadvantagesin describing the degree of future risks. Value at Risk(VaR) not only easily computed,but also easily accepted, now is developing quickly. Though some scholars proposed thatValue at Risk(VaR) doesn't satisfy sub-additivity in some models and have proposed somenew risk measures. In the case of normal distribution when loss probability is less than0.5, it is confirmed that VaR satisfies sub-additivity, so it is a coherent risk measure, whileloss probability is larger than 0.5, VaR is not a coherent risk measure. But in practicalapplications people only concern about the case when loss probability is very small, so thework about VaR is still valuable. It is known to us that VaR has closely relationship withquantile estimator, thus we can first get down to discussing quantile estimator, and furtherget the statistical properties of VaR.In fact, scholars have done lots of work about quantile estimators, various forms of esti-mators have discussed under independent conditions. But lots of work show that for a hugenumber of financial and economic time series, independence doesn't hold, and the depen-dence is an intrinsic feature. It can be note that it is not enough to discuss the estimatorsonly under independent conditions, so it has important application value to consider variousforms of estimators under dependent conditions. Generally, considering the statistical prop-erties of order statistic and some quantiles which are formed by order statistic, we alwaysfirst give the Bahadur representation of those estimator , and then do some correspondingdiscussions. In this paper, we mainly discuss the Bahadur representation of kernel quantileestimators which was first proposed by Parzen[1](1979,P113). In fact, some scholars have discussedits properties under independent condition, such as Yang(1985) established the Bahadurrepresentation of kernel-type quantile estimator in senses of convergence in probability forindependent random variables, but did not give the convergence rate.Yang(1985) established the Bahadur representation of kernel-type quantile estimator insenses of convergence in probability for independent random variables, but did not give theconvergence rate. In the paper, we extend the result of Yang[2](1985) to the case ofα-mixingrandom variable sequence, establish the Bahadur representation of VaR estimation in sensesof almost sure convergence and get the convergence rate. Further, we prove the strong con-sistence of the nonparametric VaR estimator and its convergence rate is O(n?21(log log n)21)by using Bahadur representation. Meanwhile, we established the asymptotic normality ofthe nonparametric VaR estimator and get asymptotic confidence intervals in 99% confidencelevel of VaR. Finally, we use four common time-series models to evaluate the performance ofthe nonparametric VaR estimator and verify that the probability level impact the degree ofthe estimation precision of VaR estimator. Meanwhile, we further do some empirical study.In this section, we apply the proposed kernel estimator to estimate the VaR of three financialtime series and do some comparisons.The paper is structured as follows.In chapter 1, some knowledge about VaR and research about kernel quantile estimatorare first introduced .In chapter 2, the concept ofα-mixing and some basic assumptions are given, meanwhile,the main results in the paper and some remarks about the results are showed.In chapter 3, several lemmas which are needed in the paper are given, some are thework of predecessors, some are proved in the paper based on the past work . For example, inlemma2.5 and lemma2.12. In lemma2.5, we prove the strong consistence of sample quantileby using the iterated logarithm of quantile process, the method gets rid of the common wayto prove the strong consistence. Meanwhile, In lemma2.12, we also get the strong consistenceof Kiefer process when the first parameterλis given.In chapter 4, the part is the main proof process, we prove three results in detail anddo some simulation. In Theorem1, we get the Bahadur representation of Tn(λ) in senses ofalmost sure convergence underα-mixing random variable sequence by using the propertiesof general quantile process and Kiefer process, it extend the result of Yang[2](1985) and getits convergence rate. Theorem2 and Theorem 3 are all gotten based on Theorem1. In The-orem2, we proved strong consistence of the nonparametric VaR estimator by using iteratedlogarithm of part sum random variable series underα-mixing and get the convergence rateis O(n?21(log log n)21). In Theorem3, we prove the asymptotic normality of the nonparamet- ric VaR estimator by using the method of block and get asymptotic confidence intervals in99% confidence level of VaR. The results of simulation note that the nonparametric VaRestimator have better performance than sample quantile estimator. In the empirical study,we apply the the proposed kernel estimator to estimate the VaR of three financial time se-ries, three financial time series are China Merchants Bank, Pingan Insurance of China andShangzheng composite index. Results show that the risk estimator of China Merchants Bankis the least, Shangzheng composite index is the less and then is Pingan Insurance of China.So, we consider that the investment in Pingan of China is more risky .In chapter 5, we summarize the main results of this paper and propose that the resultscan be extended to other dependence conditions. |
