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Asymptotics for the linear kernel quantile estimator. (English) Zbl 1439.62119

Summary: The method of linear kernel quantile estimator was proposed by E. Parzen [J. Am. Stat. Assoc. 74, 105–122 (1979; Zbl 0407.62001)], 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.

MSC:

62G30 Order statistics; empirical distribution functions
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics
91B84 Economic time series analysis

Citations:

Zbl 0407.62001
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References:

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