Monte Carlo pp 255-334 | Cite as

Increasing Efficiency

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)

Abstract

Section 2.7 broadened the concept of a Monte Carlo experiment to the general problem of evaluating the Lebesgue-Stieltjes integral
$$\zeta = \int_\xi {k\left( z \right)} dF\left( z \right),$$
(1)
where z = (z 1,..., z m ), {F(z)} is a joint d.f. on the m-dimensional region and {k(z)} denotes a weighting kernel defined on Also, the introduction to Ch. 3 indicates that alternative d.f.s and kernels may exist which satisfy expression (1). We call each {F(z), k(z);z } satisfying expression (1) a sampling plan, since each provides a basis for generating data which can be used to estimate ζ unbiasedly. Whereas Ch. 3 describes how to generate data efficiently from a particular {F(z), z } once a sampling plan is chosen, the present chapter studies the relative desirability of alternative sampling plans from the viewpoint of computational efficiency.

Keywords

Sampling Plan Unbiased Estimator Importance Sampling Variance Reduction Percent Confidence Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arnold, H.J., B.D. Bucher, H.F. Trotter, and J.W. Tukey (1956). Monte Carlo techniques in a complex problem about normal sampling, Symposium on Monte Carlo Methods, H.A. Meyer ed., Wiley, New York, pp. 80–84.Google Scholar
  2. Alexopoulos, C. (1991). Private communication.Google Scholar
  3. Avramidis, A.N. and J.R. Wilson (1990). Control variates for stochastic network simulation, Proc. 1990 Winter Simulation Conference, O. Balci, R.P. Sadowski, and R.E. Nance, eds., Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 323–332.Google Scholar
  4. Bahadur, R.R. and R. Ranga Rao (1960). On deviations of the sample mean, Ann. Math. Statist., 31, 1015–1027.CrossRefGoogle Scholar
  5. Bauer, K.W., Jr., and J.R. Wilson (1989). Control-variate selection criteria, SMS 89–13, School of Industrial Engineering, Purdue University, West Lafayette, IN.Google Scholar
  6. Bauer, K.W., Jr., S. Venkatraman, and J.R. Wilson (1987). Estimation procedures based on control variates with known covariance matrix, Proceedings of the 1987 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 334–341.Google Scholar
  7. Bucklew, J.A. (1989). Quick simulation and Markov chains, Department of Electrical and Computer Engineering, University of Wisconsin-Madison.Google Scholar
  8. Cheng, R.C.H. (1978). Analysis of simulation experiments under normality assumptions, J. Operational Research Society, 29, 493–497.Google Scholar
  9. Cheng, R.C.H. and T. Davenport (1989). The problem of dimensionality in stratified sampling, Management Science, 35, 1278–1296.CrossRefGoogle Scholar
  10. Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations, Ann. Math. Statist., 23, 493–507.CrossRefGoogle Scholar
  11. Chow, Y.S., H. Robbins, and D. Siegmund (1971). Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston, MA.Google Scholar
  12. Cramér, H. (1938). Sur un nouveau théorème-limite de la théorie des probabilités, Actualités Scientifiques et Industrielles, no. 736, Hermann Cie, Paris.Google Scholar
  13. Fishman, G.S. (1989). Monte Carlo, control variates, and stochastic ordering, SIAM J. Science Stat. Comp., 10, 187–204.CrossRefGoogle Scholar
  14. Fishman, G.S. and B. Huang (1983). Antithetic variates revisited, Comm. ACM, 26, 964971.Google Scholar
  15. Ford, L.R., Jr. and D.R. Fulkerson (1962). Flows in Networks, Princeton University Press, Princeton, NJ.Google Scholar
  16. Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données, Ann. Univ. Lyon Sect. A, 14, 53–77.Google Scholar
  17. Frigessi, A. and C. Vercellis (1984). An analysis of Monte Carlo algorithms for counting problems, IAMI-84. 2, Department of Mathematics, University of Milan.Google Scholar
  18. Granovsky, B.L. (1981). Optimal formulae of the conditional Monte Carlo, SIAM J. Alg. and Dis. Methods, 2, 289–294.CrossRefGoogle Scholar
  19. Granovsky, B.L. (1983). Optimal variance reduction theorem in simulation by the Monte Carlo method, Sitzungsberichte Österreichische Akademie der Wissenschaften, Mathematisch Naturwissenschaftliche Klasse, Band 192, Heft 8–10.Google Scholar
  20. Hammersley, J.M. (1956). Conditional Monte Carlo, J. ACM, 3, 73–76.CrossRefGoogle Scholar
  21. Hammersley, J.M. and D.C. Handscomb (1964). Monte Carlo Methods, Methuen, London.CrossRefGoogle Scholar
  22. Hammersley, J.M. and J.G. Mauldon (1956). General principles of antithetic variates, Proc. Comb. Phil. Soc., 52, 476–481.CrossRefGoogle Scholar
  23. Hammersley, J.M. and K.W. Morton (1956). A new Monte Carlo technique: antithetic variates, Proc. Comb. Phil. Soc., 52, 449–475.CrossRefGoogle Scholar
  24. Handscomb, H.D. (1958). Proof of the antithetic-variates theorem for n > 2, Proc. Comb. Philos. Soc., 54, 300–301.CrossRefGoogle Scholar
  25. Hoeffding, W. (1940). Masstabvinvariante Korrelationstheorie, Schriften des Mathematischen Instituts für Angewandte Mathematik der Universität Berlin 5, 179–233.Google Scholar
  26. Jun, C.H. and S.M. Ross (1992). System reliability by simulation: random hazards versus importance sampling, Probability in the Engineering and Informational Sciences, 6, 119–126.CrossRefGoogle Scholar
  27. Kahn, H. (1950). Modification of the Monte Carlo method, Seminar on Scientific Computations, C.C. Hurd ed., November 16–18, 1949, IBM.Google Scholar
  28. Kantorovich, L.V. (1948). Functional Analysis and Applied Mathematics, Volume 3, Uspehi Mat. Nauka, Russia, pp. 89–185; translated from the Russian by C.D. Benster, National Bureau of Standards, Report no. 1509, March 1952.Google Scholar
  29. Karlin, S. and H. Taylor (1975). A First Course in Stochastic Processes,2nd ed., Holden-Day.Google Scholar
  30. Kelly, F.P. (1979). Reversibility and Stochastic Networks, Wiley, New York.Google Scholar
  31. Lavenberg, S.S. and P.D. Welch (1981). A perspective on the use of control variates to increase the efficiency of Monte Carlo simulation, Man. Sci., 27, 322–335.CrossRefGoogle Scholar
  32. Lavenberg, S.S., T.L. Moeller and P.D. Welch (1982). Statistical results on control variables with application to queueing network simulation, Oper. Res., 30, 182–202.CrossRefGoogle Scholar
  33. Lehmann, E.L. (1966). Some concepts of dependence, Ann. Math. Statist., 37, 1137–1153.CrossRefGoogle Scholar
  34. Nelson, B.L. (1988). Control-variate remedies, Working Paper Series No. 1988–004, Department of Industrial and Systems Engineering, The Ohio State University.Google Scholar
  35. Porta Nova, A.M. and J.R. Wilson (1989). Estimation of multiresponse simulation meta-models using control variates, Management Science, 35, 1316–1333.CrossRefGoogle Scholar
  36. Rubinstein, R.Y. and R. Markus (1985). Efficiency of multivariate control variates in Monte Carlo simulation, Operations Research, 33, 661–677.CrossRefGoogle Scholar
  37. Rubinstein, R.Y. and G. Samarodnitsky (1987). A modified version of Handscomb’s antithetic variates theorem, SIAM J. Stat. Comp. Simul., 8, 82–98.CrossRefGoogle Scholar
  38. Sadowsky, J.S. and J.A. Bucklew (1989). Large deviations theory techniques in Monte Carlo Simulations, Proceedings of the 1989 Winter Simulation Conference, F.A. MacNair, K.J. Musselman, and P. Heidelberger eds., Institute of Electrical and Electronics Engineers, Piscataway, NJ.Google Scholar
  39. Siegmund, D. (1975). A note on the error probabilities and average sample number of the sequential probability ratio test, J. Roy. Statistical Society, Series B, 37, 394–401.Google Scholar
  40. Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests, Ann. Statist., 4, 673–684.CrossRefGoogle Scholar
  41. Trotter, H.F. and J.W. Tukey (1956). Conditional Monte Carlo for normal samples, Symposium on Monte Carlo Methods, H.A. Meyer ed., Wiley, New York, pp. 64–79.Google Scholar
  42. Venkatraman, S. and J.R. Wilson (1986). The efficiency of control variates in multiresponse simulation, Operations Research Letters, 5, 37–42.CrossRefGoogle Scholar
  43. Wald, A. (1947). Sequential Analysis, Wiley, New York.Google Scholar
  44. Walrand, J. (1988). An Introduction to Queueing Networks, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  45. Wendell, J.G. (1957). Groups and conditional Monte Carlo, Ann. Math. Statist., 28, 1048–1052.CrossRefGoogle Scholar
  46. Wilson, J.R. (1979). Proof of the antithetic-variate theorem for unbounded functions, Math. Proc. Comb. Phil. Soc., 86, 477–479.CrossRefGoogle Scholar
  47. Wilson, J.R. (1983). Antithetic sampling with multivariate inputs, Amer. J. Math. and Man. Sc., 3, 121–144.Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

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