Advertisement

Statistics and Computing

, Volume 27, Issue 2, pp 439–448 | Cite as

A Laplace transform inversion method for probability distribution functions

  • Stephen G. WalkerEmail author
Article

Abstract

This paper introduces a new Laplace transform inversion method designed specifically for when the target function is a probability distribution function. In particular, we use fixed point theory and Mann type iterative algorithms to provide a means by which to estimate and sample from the target probability distribution.

Keywords

Recursive estimation Fixed point solution 

Notes

Acknowledgments

I am very grateful for the comments and suggestions of two anonymous referees on an earlier version of the paper.

References

  1. Abate, J., Whitt, W.: Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 38–43 (1995)CrossRefzbMATHGoogle Scholar
  2. Abate, J., Choudhury, G.L., Whitt, W.: An introduction to numerical transform inversion and its application to probability models. In: Grassmann, W. (ed.) Computational Probability, pp. 257–323. Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  3. Abate, J., Whitt, W.: A unified framework for numerically inverting Laplace transfroms. INFORMS J. Comput. 18, 408–421 (2006)CrossRefzbMATHGoogle Scholar
  4. Berberan-Santos, M.N.: Analytical inversion of the Laplace transform without contour integration: application to luminesence decay laws and other relaxation functions. J. Math. Chem. 38, 165–173 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Billingsley, P.: Convergence of Probability Measures. Wiley, Chicago (1999)CrossRefzbMATHGoogle Scholar
  6. Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)zbMATHGoogle Scholar
  7. Davis, B.: Integral Transforms and their Applications. Springer, New York (2005)Google Scholar
  8. Devroye, L.: On the computer generation of random variables with a given characteristic function. Comput. Math. Appl. 7, 547–552 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Devroye, L.: An automatic method for generating random variables with a given characteristic function. SIAM J. Appl. Math. 46, 698–719 (1986)CrossRefzbMATHGoogle Scholar
  10. Devroye, L.: Non-uniform Random Variate Generation. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  11. Duan, H., Li, G.: Random Mann iteration scheme and random fixed point theorems. Appl. Math. Lett. 18, 109–115 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ghosh, J.K., Tokdar, S.T.: Convergence and consistency of Newton’s algorithm for estimating a mixing distribution. In: Fan, J., Koul, H.L. (eds.) Frontiers in Statistics, pp. 429–443. Imperial College Press, London (2005)Google Scholar
  13. Jewell, N.P.: Mixtures of exponential distributions. Ann. Stat. 10, 479–484 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kuhlman, K.L.: Review of inverse Laplace transform algorithms for Laplace-space numerical approaches. Numer. Algorithms 63, 339–355 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kuznetsov, A.: On the convergence of the Gaver–Stehfest algorithm. SIAM J. Numer. Anal 51, 2984–2998 (2103)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mann, W.R.: Mean value methods in iterations. Proc. Am. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Martin, R., Ghosh, J.K.: Stochastic approximation and Newton’s estimate of a mixing distribution. Stat. Sci. 23, 365–382 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Martin, R., Tokdar, S.T.: Asymptotic properties of predictive recursion: robustness and rate of convergence. Electron. J. Stat. 3, 1455–1472 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Newton, M.A.: On a nonparametric recursive estimator of the mixing distribution. Sankhya Ser. A 64, 306–322 (2002)MathSciNetzbMATHGoogle Scholar
  20. Papoulis, A.: A new method of inversion of the Laplace transform. Q. Appl. Math. 14, 405–414 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ridout, M.S.: Generating random numbers from a distribution specified by its Laplace transform. Stat. Comput. 19, 439–450 (2009)Google Scholar
  22. Sakurai, T.: Numerical inversion for Laplace transforms of functions with discontinuities. Adv. Appl. Probab. 36, 616–642 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Talbot, A.: The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23, 97–120 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Tokdar, S.T., Martin, R., Ghosh, J.K.: Conistency of a recursive estimate of mixing distributions. Ann. Stat. 37, 2502–2522 (2009)CrossRefzbMATHGoogle Scholar
  25. Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasAustinUSA

Personalised recommendations