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A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions

Author

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  • Lorenzo Cappello

    (Universita Commerciale Luigi Bocconi Via Roentgen 1)

  • Stephen G. Walker

    (University of Texas at Austin)

Abstract

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

Suggested Citation

  • Lorenzo Cappello & Stephen G. Walker, 2018. "A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 777-797, June.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:2:d:10.1007_s11009-017-9587-y
    DOI: 10.1007/s11009-017-9587-y
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    References listed on IDEAS

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    1. D. P. Gaver, 1966. "Observing Stochastic Processes, and Approximate Transform Inversion," Operations Research, INFORMS, vol. 14(3), pages 444-459, June.
    2. Grübel, Rudolf & Hermesmeier, Renate, 1999. "Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 197-214, November.
    3. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    4. Mnatsakanov, Robert M., 2011. "Moment-recovered approximations of multivariate distributions: The Laplace transform inversion," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 1-7, January.
    5. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "On the Laplace Transform of the Lognormal Distribution," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 441-458, June.
    6. Ning Cai & Steven Kou, 2012. "Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model," Operations Research, INFORMS, vol. 60(1), pages 64-77, February.
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    Cited by:

    1. Sandra Fortini & Sonia Petrone, 2020. "Quasi‐Bayes properties of a procedure for sequential learning in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 1087-1114, September.

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