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Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums

Author

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  • Kemal Dinçer Dingeç

    (Gebze Technical University)

  • Wolfgang Hörmann

    (Boğaziçi University)

Abstract

For the estimation of left and right-tail probabilities and the pdf of the sum of exchangeable lognormal random vectors a new conditional Monte Carlo (CMC) algorithm is developed. It removes the randomness of the sum of all input variables and is simple and fast. For estimating the left-tail probabilities the CMC algorithm is logarithmically efficient. A further improvement of the algorithm by removing also the randomness of the radius of the normal input using close to optimal one dimensional importance sampling, results in the CMC.RCMC algorithm. For the sum of independent and identically distributed (i.i.d. ) and exchangeable lognormal vectors it is the first algorithm that has bounded relative error for the left-tail probabilities. The CMC.RCMC algorithm is logarithmically efficient for the right-tail probabilities. Numerical experiments verify that it has a very good performance for all left-tail estimation problems and a good performance for the right tail for probabilities not smaller than $$10^{-10}$$ 10 - 10 . When estimating the pdf the relative errors observed are all very close to those of the corresponding probability estimates.

Suggested Citation

  • Kemal Dinçer Dingeç & Wolfgang Hörmann, 2022. "Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2093-2121, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09899-x
    DOI: 10.1007/s11009-021-09899-x
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    References listed on IDEAS

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    1. Asmussen, Søren, 2018. "Conditional Monte Carlo for sums, with applications to insurance and finance," Annals of Actuarial Science, Cambridge University Press, vol. 12(2), pages 455-478, September.
    2. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    3. Zdravko I. Botev & Robert Salomone & Daniel Mackinlay, 2019. "Fast and accurate computation of the distribution of sums of dependent log-normals," Annals of Operations Research, Springer, vol. 280(1), pages 19-46, September.
    4. Sak, Halis & Hörmann, Wolfgang & Leydold, Josef, 2010. "Efficient risk simulations for linear asset portfolios in the t-copula model," European Journal of Operational Research, Elsevier, vol. 202(3), pages 802-809, May.
    5. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "Exponential Family Techniques for the Lognormal Left Tail," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 774-787, September.
    6. Søren Asmussen & José Blanchet & Sandeep Juneja & Leonardo Rojas-Nandayapa, 2011. "Efficient simulation of tail probabilities of sums of correlated lognormals," Annals of Operations Research, Springer, vol. 189(1), pages 5-23, September.
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