Bahadur representation of linear kernel quantile estimator of VaR under α-mixing assumptions

In this paper, it is illustrated that the linear kernel quantile estimator proposed by Parzen (1979) is a reasonable estimator for VaR. Note that Yang (1985) established a Bahadur representation of the estimator in senses of convergence in probability for independent random variables. We extend the result to the case of α-mixing random variable sequence, and it is in senses of almost surely convergence with the rate log−τn. Moreover, we get the strong consistence of the VaR estimator and its convergence rate, and mean square error of the estimator.