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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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    1. Sen, Pranab Kumar, 1972. "On the Bahadur representation of sample quantiles for sequences of [phi]-mixing random variables," Journal of Multivariate Analysis, Elsevier, vol. 2(1), pages 77-95, March.
    2. Jeremy Berkowitz & James O'Brien, 2002. "How Accurate Are Value‐at‐Risk Models at Commercial Banks?," Journal of Finance, American Finance Association, vol. 57(3), pages 1093-1111, June.
    3. Gourieroux, C. & Laurent, J. P. & Scaillet, O., 2000. "Sensitivity analysis of Values at Risk," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 225-245, November.
    4. Song Xi Chen, 2005. "Nonparametric Inference of Value-at-Risk for Dependent Financial Returns," Journal of Financial Econometrics, Oxford University Press, vol. 3(2), pages 227-255.
    5. Xiaoqin Li & Wenzhi Yang & Shuhe Hu & Xuejun Wang, 2011. "The Bahadur representation for sample quantile under NOD sequence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 59-65.
    6. Babu, Gutti Jogesh & Singh, Kesar, 1978. "On deviations between empirical and quantile processes for mixing random variables," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 532-549, December.
    7. Lihong Wang, 2010. "Kernel type smoothed quantile estimation under long memory," Statistical Papers, Springer, vol. 51(1), pages 57-67, January.
    8. Ling, Nengxiang, 2008. "The Bahadur representation for sample quantiles under negatively associated sequence," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2660-2663, November.
    9. Manuel Ordóñez Cabrera & Andrew Rosalsky & Andrei Volodin, 2012. "Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 369-385, June.
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    Cited by:

    1. Wilson Calmon & Eduardo Ferioli & Davi Lettieri & Johann Soares & Adrian Pizzinga, 2021. "An Extensive Comparison of Some Well‐Established Value at Risk Methods," International Statistical Review, International Statistical Institute, vol. 89(1), pages 148-166, April.

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