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Nelson-Aalen Tail Product-limit Process and Extreme Value Index Estimation Under Random Censorship

Author

Listed:
  • Djamel Meraghni

    (Mohamed Khider University)

  • Abdelhakim Necir

    (Mohamed Khider University)

  • Louiza Soltane

    (Mohamed Khider University)

Abstract

On the basis of Nelson-Aalen nonparametric estimator of the cumulative distribution function, we provide a weak approximation to tail product-limit process for randomly right-censored heavy-tailed data. In this context, a new consistent estimator of the extreme value index is introduced and its asymptotic normality is established only by assuming the second-order condition of regular variation of the underlying distribution tail. In addition, an estimation procedure is described for high quantiles related to the above-mentioned tail index estimator. Finally, a simulation study is carried out to evaluate the performances of the newly proposed estimators with comparison to already existing ones.

Suggested Citation

  • Djamel Meraghni & Abdelhakim Necir & Louiza Soltane, 2025. "Nelson-Aalen Tail Product-limit Process and Extreme Value Index Estimation Under Random Censorship," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 87(2), pages 526-574, August.
    • Handle: RePEc:spr:sankha:v:87:y:2025:i:2:d:10.1007_s13171-025-00384-y
    DOI: 10.1007/s13171-025-00384-y
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    References listed on IDEAS

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    1. Reynkens, Tom & Verbelen, Roel & Beirlant, Jan & Antonio, Katrien, 2017. "Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 65-77.
    2. Laurent Gardes & Gilles Stupfler, 2015. "Erratum to: Estimating extreme quantiles under random truncation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 228-228, June.
    3. Ameraoui, Abdelkader & Boukhetala, Kamal & Dupuy, Jean-François, 2016. "Bayesian estimation of the tail index of a heavy tailed distribution under random censoring," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 148-168.
    4. Laurent Gardes & Gilles Stupfler, 2015. "Estimating extreme quantiles under random truncation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 207-227, June.
    5. Peng, L., 1998. "Asymptotically unbiased estimators for the extreme-value index," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 107-115, June.
    6. Beirlant, J. & Maribe, G. & Verster, A., 2018. "Penalized bias reduction in extreme value estimation for censored Pareto-type data, and long-tailed insurance applications," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 114-122.
    7. Beirlant, J. & Bardoutsos, A. & de Wet, T. & Gijbels, I., 2016. "Bias reduced tail estimation for censored Pareto type distributions," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 78-88.
    8. Einmahl, J.H.J. & Deheuvels, P, 1996. "On the strong limiting behavior of local functionals of empirical processes based upon censored data," Other publications TiSEM eac4a4cd-81ee-4107-8c70-a, Tilburg University, School of Economics and Management.
    9. Neves, Claudia & Fraga Alves, M. I., 2004. "Reiss and Thomas' automatic selection of the number of extremes," Computational Statistics & Data Analysis, Elsevier, vol. 47(4), pages 689-704, November.
    10. Einmahl, J.H.J. & Koning, A.J., 1992. "Limit theorems for a general weighted process under random censoring," Other publications TiSEM ab26769f-cec3-4b07-9e8a-9, Tilburg University, School of Economics and Management.
    11. Laurent Gardes & Gilles Stupfler, 2015. "Erratum to: Estimating extreme quantiles under random truncation," Post-Print hal-01456099, HAL.
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