Asymptotics for the linear kernel quantile estimator

Abstract

The method of linear kernel quantile estimator was proposed by Parzen (J Am Stat Assoc 74:105–121, 1979), which is a reasonable estimator for Value-at-risk (VaR). In this paper, we mainly investigate the asymptotic properties for linear kernel quantile estimator of VaR based on $$\varphi $$-mixing samples. At first, the Bahadur representation for sample quantiles under $$\varphi $$-mixing sequence is established. By using the Bahadur representation for sample quantiles, we further obtain the Bahadur representation for linear kernel quantile estimator of VaR in sense of almost surely convergence with the rate $$O\left( n^{-1/2}\log ^{-\alpha }n\right) $$ for some $$\alpha >0$$. In addition, the strong consistency for the linear kernel quantile estimator of VaR with the convergence rate $$O\left( n^{-1/2}(\log \log n)^{1/2}\right) $$ is established, and the asymptotic normality for linear kernel quantile estimator of VaR based on $$\varphi $$-mixing samples is obtained. Finally, a simulation study and a real data analysis are undertaken to assess the finite sample performance of the results that we established.