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Local polynomial maximum likelihood estimation for Pareto-type distributions

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  • Beirlant, Jan
  • Goegebeur, Yuri

Abstract

We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples.

Suggested Citation

  • Beirlant, Jan & Goegebeur, Yuri, 2004. "Local polynomial maximum likelihood estimation for Pareto-type distributions," Journal of Multivariate Analysis, Elsevier, vol. 89(1), pages 97-118, April.
    • Handle: RePEc:eee:jmvana:v:89:y:2004:i:1:p:97-118
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    References listed on IDEAS

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    1. Beirlant, Jan & Goegebeur, Yuri, 2003. "Regression with response distributions of Pareto-type," Computational Statistics & Data Analysis, Elsevier, vol. 42(4), pages 595-619, April.
    2. A. C. Davison & N. I. Ramesh, 2000. "Local likelihood smoothing of sample extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 191-208.
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    1. Farkas, Sébastien & Lopez, Olivier & Thomas, Maud, 2021. "Cyber claim analysis using Generalized Pareto regression trees with applications to insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 92-105.
    2. Bousebata, Meryem & Enjolras, Geoffroy & Girard, Stéphane, 2023. "Extreme partial least-squares," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    3. Gardes, Laurent & Girard, Stéphane, 2008. "A moving window approach for nonparametric estimation of the conditional tail index," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2368-2388, November.
    4. Daouia, Abdelaati & Gardes, Laurent & Girard, Stephane, 2011. "On kernel smoothing for extremal quantile regression," LIDAM Discussion Papers ISBA 2011031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Yaolan Ma & Bo Wei & Wei Huang, 2020. "A nonparametric estimator for the conditional tail index of Pareto-type distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(1), pages 17-44, January.
    6. Fontanari, Andrea & Cirillo, Pasquale & Oosterlee, Cornelis W., 2018. "From Concentration Profiles to Concentration Maps. New tools for the study of loss distributions," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 13-29.
    7. Goedele Dierckx & Yuri Goegebeur & Armelle Guillou, 2014. "Local robust and asymptotically unbiased estimation of conditional Pareto-type tails," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 330-355, June.
    8. Gardes, Laurent & Girard, Stéphane & Lekina, Alexandre, 2010. "Functional nonparametric estimation of conditional extreme quantiles," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 419-433, February.
    9. Ma, Yaolan & Jiang, Yuexiang & Huang, Wei, 2018. "Empirical likelihood based inference for conditional Pareto-type tail index," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 114-121.
    10. Takuma Yoshida, 2021. "Additive models for extremal quantile regression with Pareto-type distributions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(1), pages 103-134, March.

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