Introduction
As explained in the entry Uniform Random Number Generators, the simulation of random variables on a computer operates in two steps: In the first step, uniform random number generators produce imitations of i.i.d. \(U(0,1)\) (uniform over (0,1)) random variables, and in the second step these numbers are transformed in an appropriate way to imitate random variables from other distributions than the uniform ones, and other types of random objects. Here we discuss the second step only, assuming that infinite sequences of (truly) i.i.d. \(U(0,1)\) random variables are available from the first step. This assumption is not realized exactly in software implementations, but good-enough approximations are available (L’Ecuyer 2004).
For some distributions, simple exact transformations from the uniform to the target distribution are available, usually based on the inversion method. But for many types of distributions and processes, in particular those having shape parameters, and...
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References and Further Reading
Ahrens JH, Kohrt KD (1981) Computer methods for efficient sampling from largely arbitrary statistical distributions. Computing 26:19–31
Asmussen S, Glynn PW (2007) Stochastic simulation. Springer, New York
Bertoin J (1996) Lévy Processes. Cambridge University Press, Cambridge
Blair JM, Edwards CA, Johnson JH (1976) Rational Chebyshev approximations for the inverse of the error function. Math Comput 30:827–830
Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall, Englewood Cliffs
Chen HC, Asau Y (1974) On generating random variates from an empirical distribution. AIEE Trans 6:163–166
Derflinger G, Hörmann W, Leydold J (2010) Random variate generation by numerical inversion when only the density is known. ACM Trans Model Comput Simul 20(4)
Devroye L (1986) Non-Uniform random variate generation. Springer, New York
Devroye L (2006) Nonuniform random variate generation. In: Henderson SG, Nelson BL(eds) Simulation, handbooks in operations research and management science, Chap. 4. Elsevier, Amsterdam, pp 83–121
Glasserman P (2004) Monte Carlo methods in financial engineering. Springer, New York
Hörmann W, Derflinger G (2002) Fast generation of order statistics. ACM Trans Model Comput Simul 12(2):83–93
Hörmann W, Leydold J (2003) Continuous random variate generation by fast numerical inversion. ACM Trans Model Comput Simul 13(4):347–362
Hörmann W, Leydold J, Derflinger G (2004) Automatic nonuniform random variate generation. Springer, Berlin
Kinderman AJ, Monahan JF (1977) Computer generation of random variables using the ratio of uniform deviates. ACM Trans Math Softw 3:257–260
L’Ecuyer P (2004) Random number generation. In: Gentle JE, Haerdle W, Mori Y (eds) Handbook of computational statistics, Chap. II.2, Springer, Berlin, pp 35–70
L’Ecuyer P (2008) SSJ: a java library for stochastic simulation. Software user’s guide. http://www.iro.umontreal.ca/~lecuyer
L’Ecuyer P (2009) Quasi-Monte Carlo methods with applications in finance. Finance Stochastics 13(3):307–349
Law AM, Kelton WD (2000) Simulation modeling and analysis, 3rd edn. McGraw-Hill, New York
Nelsen RB (1999) An introduction to Copulas. Lecture Notes in Statistics, vol 139. Springer, New York
von Neumann J (1951) Various techniques used in connection with random digits. In: Householder As et al (ed) The Monte Carlo method, vol 12. National Bureau of Standards Applied Mathematics Series, Washington, pp 36–38
Walker AJ (1974) New fast method for generating discrete random numbers with arbitrary frequency distributions. Electron Lett 10:127–128
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L’Ecuyer, P. (2011). Non-uniform Random Variate Generations. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_408
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