Accuracy in random number generation
Author:
John F. Monahan
Journal:
Math. Comp. 45 (1985), 559568
MSC:
Primary 65C10; Secondary 68U20
MathSciNet review:
804945
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Abstract: The generation of continuous random variables on a digital computer encounters a problem of accuracy caused by approximations and discretization error. These in turn impose a bias on the simulation results. An ideal discrete approximation of a continuous distribution and a measure of error are proposed. Heuristic analysis of common methods for transforming uniform deviates to other continuous random variables is discussed. Comments and recommendations are made for the design of algorithms to reduce the bias and avoid overflow problems.
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 G. Dahlquist & A. Björck, Numerical Methods (N. Anderson, transl.), PrenticeHall, Englewood Cliffs, N. J., 1974. MR 0368379 (51:4620)
 [2]
 P. J. Davis & P. Rabinowitz, Numerical Integration, Academic Press, New York, 1975. MR 0448814 (56:7119)
 [3]
 L. Devroye, "A note on approximations in random variate generation," J. Statist. Comput. Simulation, v. 14, 1982, pp. 149158. MR 651481 (83g:62035)
 [4]
 U. Dieter & J. H. Ahrens, "A combinatorial method for the generation of normally distributed random numbers," Computing, v. 11, 1973, pp. 137146. MR 0388727 (52:9561)
 [5]
 G. Forsythe, "Von Neumann's comparison method for random sampling from the normal and other distributions," Math. Comp., v. 26, 1972, pp. 817826. MR 0315863 (47:4412)
 [6]
 J. E. Gentle, "Portability considerations for random number generation," Comp. Science and Statist.: Proc. 13th Sympos. Interface (W. F. Eddy, ed.), SpringerVerlag, Berlin and New York, 1981, pp. 158164.
 [7]
 A. J. Kinderman & J. F. Monahan, "Computer generation of random variables using the ratio of uniform deviates," ACM Trans. Math. Software, v. 3, 1977, pp. 257260.
 [8]
 D. E. Knuth & A. C. Yao, "The complexity of nonuniform random number generation," in Algorithms and Complexity, New Directions and Recent Results (J. F. Traub, ed.), Academic Press, New York, 1976, pp. 357428. MR 0431601 (55:4598)
 [9]
 P. A. W. Lewis, A. S. Goodman & J. M. Miller, "A pseudorandom number generator for the System/360," IBM Systems J., v. 8, 1969, pp. 136146.
 [10]
 M. Loeve, Probability Theory, 4th ed., SpringerVerlag, Berlin and New York, 1977. MR 0651017 (58:31324a)
 [11]
 J. F. Monahan, "Extensions of Von Neumann's method for generating random variables," Math. Comp., v. 33, 1979, pp. 10651069. MR 528058 (80c:65022)
 [12]
 R. E. Odeh & J. O. Evans, "Algorithm AS 70: Percentage points of the normal distributions," Appl. Statist., v. 23, 1974, pp. 9697.
 [13]
 L. Schrage, "A more portable Fortran random number generator," ACM Trans. Math. Software, v. 5, 1979, pp. 132138.
 [14]
 J. von Neumann, "Various techniques used in connection with random digits," in Monte Carlo Method, Nat. Bur. Standards Appl. Math. Series, v. 12, 1951, pp. 3638.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819850804945X
PII:
S 00255718(1985)0804945X
Keywords:
Random number generation,
discretization error,
approximation error,
relative accuracy,
floatingpoint arithmetic
Article copyright:
© Copyright 1985
American Mathematical Society
