IDEAS home Printed from https://ideas.repec.org/a/spr/testjl/v28y2019i4d10.1007_s11749-019-00627-9.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11749-019-00627-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11749-019-00627-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. 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.
    5. Lihong Wang, 2010. "Kernel type smoothed quantile estimation under long memory," Statistical Papers, Springer, vol. 51(1), pages 57-67, January.
    6. 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.
    7. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Qinchi Zhang & Wenzhi Yang & Shuhe Hu, 2014. "On Bahadur representation for sample quantiles under α-mixing sequence," Statistical Papers, Springer, vol. 55(2), pages 285-299, May.
    2. Zhongde Luo, 2020. "Nonparametric kernel estimation of CVaR under $$\alpha $$α-mixing sequences," Statistical Papers, Springer, vol. 61(2), pages 615-643, April.
    3. Dingshi Tian & Zongwu Cai & Ying Fang, 2018. "Econometric Modeling of Risk Measures: A Selective Review of the Recent Literature," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 201807, University of Kansas, Department of Economics, revised Oct 2018.
    4. Alexander, Carol & Sheedy, Elizabeth, 2008. "Developing a stress testing framework based on market risk models," Journal of Banking & Finance, Elsevier, vol. 32(10), pages 2220-2236, October.
    5. Dagmara Dudek & Anna Kuczmaszewska, 2024. "Some practical and theoretical issues related to the quantile estimators," Statistical Papers, Springer, vol. 65(6), pages 3917-3933, August.
    6. Hussein Khraibani & Bilal Nehme & Olivier Strauss, 2018. "Interval Estimation of Value-at-Risk Based on Nonparametric Models," Econometrics, MDPI, vol. 6(4), pages 1-30, December.
    7. James Ming Chen, 2018. "On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles," Risks, MDPI, vol. 6(2), pages 1-28, June.
    8. Lu Ou & Zhibiao Zhao, 2021. "Value‐at‐risk forecasting via dynamic asymmetric exponential power distributions," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(2), pages 291-300, March.
    9. Cotter, John & Dowd, Kevin, 2007. "Evaluating the Precision of Estimators of Quantile-Based Risk Measures," MPRA Paper 3504, University Library of Munich, Germany.
    10. Biswajit Patra & Puja Padhi, 2015. "Backtesting of Value at Risk Methodology: Analysis of Banking Shares in India," Margin: The Journal of Applied Economic Research, National Council of Applied Economic Research, vol. 9(3), pages 254-277, August.
    11. Lin Fan & Peter W. Glynn & Markus Pelger, 2018. "Change-Point Testing for Risk Measures in Time Series," Papers 1809.02303, arXiv.org, revised Jul 2023.
    12. Elena-Ivona Dumitrescu & Christophe Hurlin & Vinson Pham, 2012. "Backtesting Value-at-Risk: From Dynamic Quantile to Dynamic Binary Tests," Finance, Presses universitaires de Grenoble, vol. 33(1), pages 79-112.
    13. Abdoul G. Sam, 2010. "Nonparametric estimation of market risk: an application to agricultural commodity futures," Agricultural Finance Review, Emerald Group Publishing Limited, vol. 70(2), pages 285-297, August.
    14. Wang, Chuan-Sheng & Zhao, Zhibiao, 2016. "Conditional Value-at-Risk: Semiparametric estimation and inference," Journal of Econometrics, Elsevier, vol. 195(1), pages 86-103.
    15. Ho, Hwai-Chung, 2015. "Sample quantile analysis for long-memory stochastic volatility models," Journal of Econometrics, Elsevier, vol. 189(2), pages 360-370.
    16. So Yeon Chun & Alexander Shapiro & Stan Uryasev, 2012. "Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics," Operations Research, INFORMS, vol. 60(4), pages 739-756, August.
    17. Wendler, Martin, 2011. "Bahadur representation for U-quantiles of dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1064-1079, July.
    18. Nour-Eddine Berrahou & Salim Bouzebda & Lahcen Douge, 2024. "The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-37, June.
    19. Ivanov Roman V., 2018. "On risk measuring in the variance-gamma model," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 23-33, January.
    20. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:testjl:v:28:y:2019:i:4:d:10.1007_s11749-019-00627-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.