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Approximating the Probability Density Function of a Transformation of Random Variables

Author

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  • Denys Pommeret

    (Institut de Mathématiques de Marseille - CNRS - Ecole Centrale - Case 907)

  • Laurence Reboul

    (Institut de Mathématiques de Marseille - CNRS - Ecole Centrale - Case 907)

Abstract

We propose a general formula for the probability density function of transformations of continuous or discrete random variables. Approximations and estimations are derived. Particular cases are treated when transformations are sum or products of random variables. The formula has a simple form when probability density functions are expressed with respect to a reference measure which belongs to the class of natural exponential families with quadratic variance functions. Some numerical results are provided to illustrate the method.

Suggested Citation

  • Denys Pommeret & Laurence Reboul, 2019. "Approximating the Probability Density Function of a Transformation of Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 633-645, June.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:2:d:10.1007_s11009-018-9629-0
    DOI: 10.1007/s11009-018-9629-0
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    References listed on IDEAS

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    1. Nadarajah, Saralees & Kotz, Samuel, 2006. "On The Product And Ratio Of Gamma And Weibull Random Variables," Econometric Theory, Cambridge University Press, vol. 22(2), pages 338-344, April.
    2. Anwar Joarder, 2009. "Moments of the product and ratio of two correlated chi-square variables," Statistical Papers, Springer, vol. 50(3), pages 581-592, June.
    3. Pierre-Olivier Goffard & Stéphane Loisel & Denys Pommeret, 2017. "Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 151-174, March.
    4. M. Shakil & B. Golam Kibria & Kuang-Chao Chang, 2008. "Distributions of the product and ratio of Maxwell and Rayleigh random variables," Statistical Papers, Springer, vol. 49(4), pages 729-747, October.
    5. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2013. "On the exact and approximate distributions of the product of a Wishart matrix with a normal vector," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 70-81.
    6. Asmussen, Søren & Rojas-Nandayapa, Leonardo, 2008. "Asymptotics of sums of lognormal random variables with Gaussian copula," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2709-2714, November.
    7. Christopher Withers & Saralees Nadarajah, 2013. "On the product of gamma random variables," Quality & Quantity: International Journal of Methodology, Springer, vol. 47(1), pages 545-552, January.
    8. Marques, Filipe J. & Loingeville, Florence, 2016. "Improved near-exact distributions for the product of independent Generalized Gamma random variables," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 55-66.
    9. Glickman, Theodore S. & Xu, Feng, 2008. "The distribution of the product of two triangular random variables," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2821-2826, November.
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