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Renewal sums under mixtures of exponentials

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  • Zhang, Zhehao

Abstract

We start with applying two methods to derive formulas of a mixture of exponential process, i.e., a renewal process whose inter-arrival time follows a mixture of exponentials. Further, stochastic order properties are discussed when comparing this process to a Poisson process with the same expectation of inter-arrival times. Based on these properties, formulas and ordering properties are given for the non-discounted compound process as well as the discounted one. Explicit formulas for the density functions are also provided for both cases. Under the discounted compound case, several new results are derived for heavy-tailed distributions. Finally, the Laguerre series approximation is proposed and tested for various common actuarial indices, e.g., VaR, CTE and stop-loss premium.

Suggested Citation

  • Zhang, Zhehao, 2018. "Renewal sums under mixtures of exponentials," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 281-301.
    • Handle: RePEc:eee:apmaco:v:337:y:2018:i:c:p:281-301
    DOI: 10.1016/j.amc.2018.05.031
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    References listed on IDEAS

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    1. Daniel Dufresne, 2007. "Fitting combinations of exponentials to probability distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 23-48, January.
    2. Devroye, Luc, 1989. "On random variate generation when only moments or Fourier coefficients are known," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 31(1), pages 71-89.
    3. K. Anaya-Izquierdo & P. Marriott, 2007. "Local mixtures of the exponential distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(1), pages 111-134, March.
    4. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
    5. R. Pillai, 1990. "On Mittag-Leffler functions and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 157-161, March.
    6. James Vaupel & Kenneth Manton & Eric Stallard, 1979. "The impact of heterogeneity in individual frailty on the dynamics of mortality," Demography, Springer;Population Association of America (PAA), vol. 16(3), pages 439-454, August.
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    Cited by:

    1. Okhli, Kheirolah & Jabbari Nooghabi, Mehdi, 2023. "On the three-component mixture of exponential distributions: A Bayesian framework to model data with multiple lower and upper outliers," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 480-500.
    2. Zhang, Zhehao, 2019. "On the stochastic equation L(Z)=L[V(X+Z)] and properties of Mittag–Leffler distributions," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 365-376.
    3. Okhli, Kheirolah & Jabbari Nooghabi, Mehdi, 2021. "On the contaminated exponential distribution: A theoretical Bayesian approach for modeling positive-valued insurance claim data with outliers," Applied Mathematics and Computation, Elsevier, vol. 392(C).

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