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Size-Biased Risk Measures of Compound Sums

Author

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  • Denuit, Michel

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

The size-biased, or length-biased transform is known to be particularly useful in insurance risk measurement. The case of continuous losses has been extensively considered in the actuarial literature. Given their importance in insurance studies, this article concentrates on compound sums. The zero-augmented distributions that naturally appear in the individual model of risk theory are obtained as particular cases when the claim frequency distribution is concentrated on {0, 1}. The general results derived in this article help actuaries to understand how risk measurement proceeds because the formulas make explicit the loadings corresponding to each source of randomness. Some simple and explicit expressions are obtained when losses are modeled by independent compound Poisson sums and compound mixed Poisson sums, including the compound negative binomial sums. Extensions to correlated risks are briefly discussed in the concluding section.

Suggested Citation

  • Denuit, Michel, 2020. "Size-Biased Risk Measures of Compound Sums," LIDAM Reprints ISBA 2020034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvar:2020034
    DOI: https://doi.org/10.1080/10920277.2019.1676787
    Note: In: North American Actuarial Journal - Vol. 24, no.4, p. 512-532 (2020)
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    Cited by:

    1. Denuit, Michel & Robert, Christian Y., 2021. "Polynomial series expansions and moment approximations for conditional mean risk sharing of insurance losses," LIDAM Discussion Papers ISBA 2021016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Denuit, M. & Robert, C.Y., 2020. "Ultimate behavior of conditional mean risk sharing for independent compound Panjer-Katz sums with gamma and Pareto severities," LIDAM Discussion Papers ISBA 2020014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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