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On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science

Author

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  • Emilio Gómez-Déniz

    (Department of Quantitative Methods in Economics and TiDES, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
    These authors contributed equally to this work.)

  • Enrique Calderín-Ojeda

    (Department of Economics, University of Melbourne, Melbourne, VIC 3010, Australia
    These authors contributed equally to this work.)

Abstract

This paper studies properties and applications related to the mixture of the class of distributions built by the Lehmann’s alternative (also referred to in the statistical literature as max-stable or exponentiated distribution) of the form [ G ( · ) ] λ , where λ > 0 and G ( · ) is a continuous cumulative distribution function. This mixture can be useful in economics, financial, and actuarial fields, where extreme and long tails appear in the empirical data. The special case in which G ( · ) is the Stoppa cumulative distribution function, which is a good description of the random behaviour of large losses, is studied in detail. We provide properties of this mixture, mainly related to the analysis of the tail of the distribution that makes it a candidate for fitting actuarial data with extreme observations. Inference procedures are discussed and applications to three well-known datasets are shown.

Suggested Citation

  • Emilio Gómez-Déniz & Enrique Calderín-Ojeda, 2023. "On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science," Risks, MDPI, vol. 12(1), pages 1-22, December.
    • Handle: RePEc:gam:jrisks:v:12:y:2023:i:1:p:6-:d:1309657
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    References listed on IDEAS

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    1. Frangos, Nikolaos & Karlis, Dimitris, 2004. "Modelling losses using an exponential-inverse Gaussian distribution," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 53-67, August.
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