Communications in Applied Mathematics and Computational Science

Approximation of probabilistic Laplace transforms and their inverses

Guillaume Coqueret

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Abstract

We present a method to approximate the law of positive random variables defined by their Laplace transforms. It is based on the study of the error in the Laplace domain and allows for many behaviors of the law, both at 0 and infinity. In most cases, both the Kantorovich/Wasserstein error and the Kolmogorov–Smirnov error can be accurately computed. Two detailed examples illustrate our results.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 7, Number 2 (2012), 231-246.

Dates
Received: 23 March 2012
Revised: 27 July 2012
Accepted: 16 August 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732057

Digital Object Identifier
doi:10.2140/camcos.2012.7.231

Mathematical Reviews number (MathSciNet)
MR3020215

Zentralblatt MATH identifier
1259.65206

Subjects
Primary: 65R32: Inverse problems
Secondary: 65C50: Other computational problems in probability

Keywords
approximation Laplace transform inversion completely monotone functions Kantorovich distance

Citation

Coqueret, Guillaume. Approximation of probabilistic Laplace transforms and their inverses. Commun. Appl. Math. Comput. Sci. 7 (2012), no. 2, 231--246. doi:10.2140/camcos.2012.7.231. https://projecteuclid.org/euclid.camcos/1513732057


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References

  • M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, no. 55, National Bureau of Standards, Washington, DC, 1966.
  • A. Al-Shuaibi, Inversion of the Laplace transform via Post–Widder formula, Integral Transform. Spec. Funct. 11 (2001), no. 3, 225–232.
  • N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, no. 27, Cambridge University Press, Cambridge, 1987.
  • K. F. Chen and S. L. Mei, Accelerations of Zhao's methods for the numerical inversion of Laplace transform, Int. J. Numer. Methods Biomed. Eng. 27 (2011), no. 2, 273–282. http://msp.org/idx/mr/2011k:65185MR 2011k:65185
  • A. M. Cohen, Numerical methods for Laplace transform inversion, Numerical Methods and Algorithms, no. 5, Springer, New York, 2007.
  • G. Dall'Aglio, Sugli estremi dei momenti delle funzioni di ripartizione doppia, Ann. Scuoloa Norm. Sup. Pisa $(3)$ 10 (1956), 35–74.
  • C. L. Epstein and J. Schotland, The bad truth about Laplace's transform, SIAM Rev. 50 (2008), no. 3, 504–520.
  • W. Feller, An introduction to probability theory and its applications, 2nd ed., vol. II, Wiley, New York, 1971.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier, Amsterdam, 2007.
  • C.-Y. Hu and G. D. Lin, Some inequalities for Laplace transforms, J. Math. Anal. Appl. 340 (2008), no. 1, 675–686.
  • P. J. Huber and E. M. Ronchetti, Robust statistics, 2nd ed., Wiley, Hoboken, NJ, 2009. http://msp.org/idx/mr/2010j:62004MR 2010j:62004
  • P. Jara, F. Neubrander, and K. Özer, Rational inversion of the Laplace transform, J. Evol. Equ. 12 (2012), no. 2, 435–457.
  • K. K. Jose, P. Uma, V. S. Lekshmi, and H. J. Haubold, Generalized Mittag-Leffler distributions and processes for applications in astrophysics and time series modeling, Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science (Tokyo, 2007) (H. J. Haubold and A. M. Mathai, eds.), Springer, Heidelberg, 2010, pp. 79–92.
  • F.-R. Lin and F. Liang, Application of high order numerical quadratures to numerical inversion of the Laplace transform, Adv. Comput. Math. 36 (2012), no. 2, 267–278.
  • I. M. Longman, Numerical Laplace transform inversion of a function arising in viscoelasticity, J. Comput. Phys. 10 (1972), no. 2, 224–231 (English).
  • Y. L. Luke, Error estimation in numerical inversion of Laplace transforms using Padé approximation, Journal of the Franklin Institute 305 (1978), no. 5, 259–273.
  • F. Mainardi, P. Paradisi, and R. Gorenflo, Probability distributions generated by fractional diffusion equations, Workshop on Econophysics (Budapest, 1997), 2007.
  • V. Masol and J. L. Teugels, Numerical accuracy of real inversion formulas for the Laplace transform, J. Comput. Appl. Math. 233 (2010), no. 10, 2521–2533.
  • K. Nakagawa, Tail probability and singularity of Laplace–Stieltjes transform of a heavy tailed random variable, Information Theory and Its Applications $($ISITA$)$ (Auckland, 2008), IEEE, Piscataway, NJ, 2009.
  • K. Oldham, J. Myland, and J. Spanier, An atlas of functions: with Equator, the atlas function calculator, 2nd ed., Springer, New York, 2009.
  • V. V. Petrov, Limit theorems of probability theory: sequences of independent random variables, Oxford Studies in Probability, no. 4, Clarendon/Oxford University Press, New York, 1995.
  • A. D. Polyanin and A. V. Manzhirov, Handbook of integral equations, 2nd ed., CRC, Boca Raton, FL, 2008.
  • A. Stef and G. Tenenbaum, Inversion de Laplace effective, Ann. Probab. 29 (2001), no. 1, 558–575.
  • V. K. Tuan and D. T. Duc, Convergence rate of Post–Widder approximate inversion of the Laplace transform, Vietnam J. Math. 28 (2000), no. 1, 93–96.
  • M. S. Veillette and M. S. Taqqu, A technique for computing the PDFs and CDFs of nonnegative infinitely divisible random variables, J. Appl. Probab. 48 (2011), no. 1, 217–237.
  • J. A. C. Weideman, Optimizing Talbot's contours for the inversion of the Laplace transform, SIAM J. Numer. Anal. 44 (2006), no. 6, 2342–2362.
  • H. Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, no. 17, Cambridge University Press, Cambridge, 2005.