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Weighted least squares estimators for the Parzen tail index

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.

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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.

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Correspondence to László Viharos.

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Appendix

Appendix

See Tables 1, 2 and 3.

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

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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

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

Keywords

  • Density-quantile
  • Tail exponent
  • Weighted least squares estimators

Mathematics Subject Classification

  • Primary 62G32
  • Secondary 60F05