Skip to main content
SpringerLink
Log in

Weighted least squares estimators for the Parzen tail index

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Estimation of the tail index of heavy-tailed distributions and its applications are essential in many research areas. We propose a class of weighted least squares (WLS) estimators for the Parzen tail index. Our approach is based on the method developed by Holan and McElroy (J Stat Plan Inference 140(12):3693–3708, 2010). We investigate consistency and asymptotic normality of the WLS estimators. Through a simulation study, we make a comparison with the Hill, Pickands, DEdH (Dekkers, Einmahl and de Haan) and ordinary least squares (OLS) estimators using the mean square error as criterion. The results show that in a restricted model some members of the WLS estimators are competitive with the Pickands, DEdH and OLS estimators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia of Mathematics and Its Applications, vol. 27 (Cambridge University Press, Cambridge, 1989)

    MATH  Google Scholar 

  2. C. Cheng, Uniform consistency of generalized kernel estimators of quantile density. Ann. Statist. 23(6), 2285–2291 (1995)

    Article  MathSciNet  Google Scholar 

  3. G. Ciuperca, C. Mercadier, Semi-parametric estimation for heavy tailed distributions. Extremes 13(1), 55–87 (2010)

    Article  MathSciNet  Google Scholar 

  4. A.L.M. Dekkers, J.H.J. Einmahl, L. de Haan, A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17(4), 1833–1855 (1989)

    MathSciNet  MATH  Google Scholar 

  5. B.M. Hill, A simple general approach to inference about the tail of a distribution. Ann. Stat. 3(5), 1163–1174 (1975)

    Article  MathSciNet  Google Scholar 

  6. S.H. Holan, T.S. McElroy, Tail exponent estimation via broadband log density-quantile regression. J. Stat. Plan. Inference 140(12), 3693–3708 (2010)

    Article  MathSciNet  Google Scholar 

  7. T. McElroy, C.H. Nagaraja, Tail index estimation with a fixed tuning parameter fraction. J. Stat. Plan. Inference 170, 27–45 (2016)

    Article  MathSciNet  Google Scholar 

  8. E. Parzen, Nonparametric statistical data modeling. J. Am. Stat. Assoc. 74(365), 105–131 (1979)

    Article  MathSciNet  Google Scholar 

  9. E. Parzen, Quantile probability and statistical data modeling. Stat. Sci. 19(4), 652–662 (2004)

    Article  MathSciNet  Google Scholar 

  10. J. Pickands III., Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the referee for helpful remarks and suggestions. This research was supported by Grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary.

Author information

Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    Amenah Al-Najafi & László Viharos

Authors
  1. Amenah Al-Najafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. László Viharos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to László Viharos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Tables 1, 2 and 3.

Table 1 Limiting variances for different weight functions and tail indices
Full size table
Table 2 Average simulated tail index estimates (mean) for sample size \(n=700\) and for \(L_0\equiv 1\)
Full size table
Table 3 Empirical mean square errors (MSE) of tail index estimates for sample size \(n=700\) and for \(L_0\equiv 1\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Al-Najafi, A., Viharos, L. Weighted least squares estimators for the Parzen tail index. Period Math Hung 84, 259–269 (2022). https://doi.org/10.1007/s10998-021-00403-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-021-00403-z

Keywords

Mathematics Subject Classification

Search

Navigation