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Error Bounds for Cumulative Distribution Functions of Convolutions via the Discrete Fourier Transform

Author

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  • Richard L. Warr

    (Brigham Young University)

  • Cason J. Wight

    (Brigham Young University)

Abstract

In statistical theory, convolutions are often avoided in favor of asymptotic approximation or simulation. Much of this is due to the fact that convolution is a challenging problem. With abundant computational resources, numerical convolution is a more viable option than in past decades. This paper proposes mathematical error bounds for the cumulative distribution function of the convolution of a finite number of independent univariate random variables. The discrete Fourier transform and its companion, the inverse discrete Fourier transform, are used to provide fast and easily obtainable mathematical error bounds for these convolutions. Examples and applications are provided to demonstrate a few possible uses of the error bounds.

Suggested Citation

  • Richard L. Warr & Cason J. Wight, 2020. "Error Bounds for Cumulative Distribution Functions of Convolutions via the Discrete Fourier Transform," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 881-904, September.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09739-z
    DOI: 10.1007/s11009-019-09739-z
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    References listed on IDEAS

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    1. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    2. Grübel, Rudolf & Hermesmeier, Renate, 2000. "Computation of Compound Distributions II: Discretization Errors and Richardson Extrapolation," ASTIN Bulletin, Cambridge University Press, vol. 30(2), pages 309-331, November.
    3. Ashraf Ben El‐Shanawany & Keith H. Ardron & Simon P. Walker, 2018. "Lognormal Approximations of Fault Tree Uncertainty Distributions," Risk Analysis, John Wiley & Sons, vol. 38(8), pages 1576-1584, August.
    4. Richard L. Warr, 2014. "Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 1025-1038, December.
    5. Ruckdeschel, Peter & Kohl, Matthias, 2014. "General Purpose Convolution Algorithm in S4 Classes by Means of FFT," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 59(i04).
    6. 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.
    7. Ng, ManWo & Waller, S. Travis, 2010. "A computationally efficient methodology to characterize travel time reliability using the fast Fourier transform," Transportation Research Part B: Methodological, Elsevier, vol. 44(10), pages 1202-1219, December.
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