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On numerical calculation of probabilities according to Dirichlet distribution

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  • Ashraf Gouda
  • Tamás Szántai

Abstract

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary. In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented. Copyright Springer Science+Business Media, LLC 2010

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  • Ashraf Gouda & Tamás Szántai, 2010. "On numerical calculation of probabilities according to Dirichlet distribution," Annals of Operations Research, Springer, vol. 177(1), pages 185-200, June.
    • Handle: RePEc:spr:annopr:v:177:y:2010:i:1:p:185-200:10.1007/s10479-009-0601-9
    DOI: 10.1007/s10479-009-0601-9
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    References listed on IDEAS

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    1. Chotikapanich, Duangkamon & Griffiths, William E, 2002. "Estimating Lorenz Curves Using a Dirichlet Distribution," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 290-295, April.
    2. Peter C.B. Phillips, 1988. "The Characteristic Function of the Dirichlet and Multivariate F Distributions," Cowles Foundation Discussion Papers 865, Cowles Foundation for Research in Economics, Yale University.
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    Cited by:

    1. René Henrion & Andris Möller, 2012. "A Gradient Formula for Linear Chance Constraints Under Gaussian Distribution," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 475-488, August.
    2. Martin Bod’a, 2017. "Stochastic sensitivity analysis of concentration measures," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(2), pages 441-471, June.
    3. József Bukszár & Gergely Mádi-Nagy & Tamás Szántai, 2012. "Computing bounds for the probability of the union of events by different methods," Annals of Operations Research, Springer, vol. 201(1), pages 63-81, December.

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