Methodology and Computing in Applied Probability

, Volume 16, Issue 4, pp 1025–1038 | Cite as

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

Article

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.

Keywords

Characteristic function First passage distribution Fast Fourier transform Semi-Markov process 

AMS 2000 Subject Classifications

30E10 60G10 60K15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abate J, Valkó P (2004) Multi-precision Laplace transform inversion. Int J Numer Methods Eng 60(5):979–993CrossRefMATHGoogle Scholar
  2. Abate J, Whitt W (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Syst 10:5–88MathSciNetCrossRefMATHGoogle Scholar
  3. Barbu VS, Limnios N (2008) Semi-Markov chains and hidden semi-Markov models toward applications: their use in reliability and DNA analysis. Springer, New YorkGoogle Scholar
  4. Bellman RE, Kalaba RE, Lockett J (1966) Numerical inversion of the Laplace transform. American Elsevier, New YorkMATHGoogle Scholar
  5. Briggs WL, Henson VE (1995) The DFT: an owner’s manual for the discrete fourier transform. SIAM, PhiladelphiaMATHGoogle Scholar
  6. Hughett P (1998) Error bounds for numerical inversion of a probability characteristic function. SIAM J Numer Anal 35(4):1368–1392MathSciNetCrossRefMATHGoogle Scholar
  7. Huzurbazar AV (2005) Flowgraph models for multistate time-to-event data. Wiley, New YorkMATHGoogle Scholar
  8. Lin FR, Liang F (2012) Application of high order numerical quadratures to numerical inversion of the Laplace transform. Adv Comput Math 36(2):267–278MathSciNetCrossRefMATHGoogle Scholar
  9. Nakagawa T, Osaki S (1975) The discrete Weibull distribution. IEEE Trans Reliab R-24(5):300–301CrossRefGoogle Scholar
  10. Pyke R (1961) Markov renewal processes with finitely many states. Ann Math Stat 32(4):1243–1259MathSciNetCrossRefMATHGoogle Scholar
  11. R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org. ISBN 3-900051-07-0
  12. Strawderman RL (2004) Computing tail probabilities by numerical Fourier inversion: the absolutely continuous case. Stat Sin 41:175–201MathSciNetGoogle Scholar
  13. Warr RL, Collins DH (2011) A comprehensive method for solving finite-state semi-Markov processes. Tech. Rep. LA-UR-11-01596, Los Alamos National Laboratory, Los Alamos, NMGoogle Scholar
  14. Yao DD (1985) First-passage-time moments of Markov processes. J Appl Probab 22(4):939–945MathSciNetCrossRefMATHGoogle Scholar
  15. Zhang X, Hou Z (2012) The first-passage times of phase semi-Markov processes. Stat Probab Lett 82(1):40–48. doi:10.1016/j.spl.2011.08.021 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2013

Authors and Affiliations

  1. 1.Air Force Institute of TechnologyDaytonUSA

Personalised recommendations