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Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform

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  • Richard L. Warr

    (Air Force Institute of Technology)

Abstract

First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. Finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be unstable for a general probability measure. However, we show they can be quickly and accurately calculated using the inverse discrete Fourier transform for lattice distributions. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques with two examples.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:4:d:10.1007_s11009-013-9366-3
    DOI: 10.1007/s11009-013-9366-3
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    References listed on IDEAS

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    1. Zhang, Xuan & Hou, Zhenting, 2012. "The first-passage times of phase semi-Markov processes," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 40-48.
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    Cited by:

    1. 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.
    2. Bernardo B. Andrade & Geraldo S. Souza, 2018. "Likelihood computation in the normal-gamma stochastic frontier model," Computational Statistics, Springer, vol. 33(2), pages 967-982, June.
    3. Bei Wu & Brenda Ivette Garcia Maya & Nikolaos Limnios, 2021. "Using Semi-Markov Chains to Solve Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1419-1431, December.

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