Abstract
This paper discusses simulation from an absolutely continuous distribution on the positive real line when the Laplace transform of the distribution is known but its density and distribution functions may not be available. We advocate simulation by the inversion method using a modified Newton-Raphson method, with values of the distribution and density functions obtained by numerical transform inversion. We show that this algorithm performs well in a series of increasingly complex examples. Caution is needed in some situations when the numerical Laplace transform inversion becomes unreliable. In particular the algorithm should not be used for distributions with finite range. But otherwise, except for rather pathological distributions, the approach offers a rapid way of generating random samples with minimal user effort. We contrast our approach with an alternative algorithm due to Devroye (Comput. Math. Appl. 7, 547–552, 1981).
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Abate, J., Valko, P.P.: Multi-precision Laplace transform inversion. Int. J. Numer. Methods Eng. 60, 979–993 (2004)
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)
Ahrens, J.H., Dieter, U.: Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing 12, 223–246 (1974)
Ahrens, J.H., Dieter, U.: Generating gamma variates by a modified rejection technique. Commun. ACM 25, 47–54 (1982)
Barndorff-Nielsen, O.E., Cox, D.R.: Asymptotic Techniques for Use in Statistics. Chapman and Hall, London (1989)
Brix, A.: Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Probab. 31 (1999)
Chambers, J.M., Mallows, C.L., Struck, B.W.: A method for simulating stable random variables. J. Am. Stat. Assoc. 71, 340–344 (1976)
Choudhury, G.L., Whitt, W.: Probabilistic scaling for the numerical inversion of nonprobability transforms. INFORMS J. Comput. 9, 175–184 (1997)
Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)
Devroye, L.: On the computer generation of random variables with a given characteristic function. Comput. Math. Appl. 7, 547–552 (1981)
Devroye, L.: Methods for generating random variates with Polya characteristic functions. Stat. Probab. Lett. 2, 257–261 (1984)
Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986a)
Devroye, L.: An automatic method for generating random variables with a given characteristic function. SIAM J. Appl. Math. 46, 698–719 (1986b)
Devroye, L.: Algorithms for generating discrete random variables with a given generating function or a given moment sequence. SIAM J. Sci. Stat. Comput. 12, 107–126 (1991)
Dunn, P.K., Smyth, G.K.: Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Stat. Comput. 18, 73–86 (2008)
Feller, W.G.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971)
Hofert, M.: Sampling Archimedean copulas. Comput. Stat. Data Anal. 52, 5163–5174 (2008)
Hougaard, P.: Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387–396 (1986)
Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997)
Lukacs, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)
Marshall, A.W., Olkin, I.: Families of multivariate distributions. J. Am. Stat. Assoc. 83, 834–841 (1988)
McCulloch, J.H., Panton, D.B.: Precise tabulation of the maximally-skewed stable distributions and densities. Comput. Stat. Data Anal. 23, 307–320 (1997)
McNeil, A.J.: Sampling nested Archimedean copulas. J. Stat. Comput. Simul. 78, 567–581 (2008)
Monahan, J.F.: Numerical Methods of Statistics. Cambridge University Press, Cambridge (2001)
Nelsen, R.: An Introduction to Copulas. Springer, New York (1999)
O’Cinneide, C.A.: Euler summation for Fourier series and Laplace transform inversion. Commun. Stat. Stoch. Models 13, 315–337 (1997)
Palmer, K.J., Ridout, M.S., Morgan, B.J.T.: Modelling cell generation times by using the tempered stable distribution. Appl. Stat. 57, 379–397 (2008)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran, 2nd edn. Cambridge University Press, Cambridge (1992)
Rosiński, J.: Simulation of Lévy processes. In: Ruggeri, F., Kenett, R., Faltin, F. (eds.) Encyclopedia of Statistics in Quality and Reliability. Wiley, New York (2007a)
Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677–707 (2007b)
Sakurai, T.: Numerical inversion for Laplace transforms of functions with discontinuities. Adv. Appl. Probab. 36, 616–642 (2004)
Schoutens, W.: Lévy Processes in Finance. Wiley, New York (2003)
Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numer. 16, 305–378 (2007)
Talbot, A.: The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23, 97–120 (1979)
Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006)
Tweedie, M.C.K.: An index which distinguishes between some important exponential families. In: Ghosh, J.K., Roy, J. (eds.) Statistics: Applications and New Directions: Proceedings of the Indian Statistical Institute Golden Jubilee International Conference, pp. 579–604. Indian Statistical Institute, Kolkata (1984)
Ushakov, N.G.: Selected Topics in Characteristic Functions. VSP, Utrecht (1999)
Whelan, N.: Sampling from Archimedean copulas. Quant. Finance 4, 339–352 (2004)
Wimp, J.: Sequence Transformations and Their Applications. Academic, New York (1981)
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Ridout, M.S. Generating random numbers from a distribution specified by its Laplace transform. Stat Comput 19, 439 (2009). https://doi.org/10.1007/s11222-008-9103-x
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DOI: https://doi.org/10.1007/s11222-008-9103-x