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Asymptotics for the linear kernel quantile estimator

Author

Listed:
  • Xuejun Wang

    (Anhui University)

  • Yi Wu

    (Anhui University)

  • Wei Yu

    (Anhui University)

  • Wenzhi Yang

    (Anhui University)

  • Shuhe Hu

    (Anhui University)

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 $$\varphi $$φ-mixing samples. At first, the Bahadur representation for sample quantiles under $$\varphi $$φ-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 $$O\left( n^{-1/2}\log ^{-\alpha }n\right) $$On-1/2log-αn for some $$\alpha >0$$α>0. In addition, the strong consistency for the linear kernel quantile estimator of VaR with the convergence rate $$O\left( n^{-1/2}(\log \log n)^{1/2}\right) $$On-1/2(loglogn)1/2 is established, and the asymptotic normality for linear kernel quantile estimator of VaR based on $$\varphi $$φ-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.

Suggested Citation

  • Xuejun Wang & Yi Wu & Wei Yu & Wenzhi Yang & Shuhe Hu, 2019. "Asymptotics for the linear kernel quantile estimator," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(4), pages 1144-1174, December.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:4:d:10.1007_s11749-019-00627-9
    DOI: 10.1007/s11749-019-00627-9
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    References listed on IDEAS

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