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pp 1–31 | Cite as

Asymptotics for the linear kernel quantile estimator

  • Xuejun WangEmail author
  • Yi Wu
  • Wei Yu
  • Wenzhi Yang
  • Shuhe Hu
Original Paper
  • 24 Downloads

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.

Keywords

Bahadur representation Linear kernel quantile estimator Value-at-risk Strong consistency Asymptotic normality 

Mathematics Subject Classification

62G30 62G20 62G05 

Notes

Acknowledgements

The authors are most grateful to the Editor and anonymous referees for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

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Copyright information

© Sociedad de Estadística e Investigación Operativa 2019

Authors and Affiliations

  • Xuejun Wang
    • 1
    Email author
  • Yi Wu
    • 1
  • Wei Yu
    • 1
  • Wenzhi Yang
    • 1
  • Shuhe Hu
    • 1
  1. 1.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China

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