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Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences

  • Art B. Owen
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 106)

Abstract

This article presents a hybrid of Monte Carlo and Quasi-Monte Carlo methods. In this hybrid, certain low discrepancy point sets and sequences due to Faure, Niederreiter and Sobol’ are obtained and their digits are randomly permuted. Since this randomization preserves the equidistribution properties of the points it also preserves the proven bounds on their quadrature errors. The accuracy of an estimated integrand can be assessed by replication, consisting of independent re-randomizations.

Keywords

Orthogonal Array Latin Hypercube Sample Probability Zero Monte Carlo Variance Elementary 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. Braaten, E. & Weller, G. (1979). An Improved Low-Discrepancy Sequence for Multidimensional Quasi-Monte Carlo Integration. J. Comp. Physics 33, 249 – 258.zbMATHCrossRefGoogle Scholar
  2. Chung, K.L. (1974). A Course in Probability Theory. Academic Press, New York.zbMATHGoogle Scholar
  3. Cranley, R. CC Patterson, T.N.L. (1976). Randomization of Number Theoretic Methods for Multiple Integration. SIAM J. Numer. Anal 13, 904 – 914.MathSciNetzbMATHGoogle Scholar
  4. Davis, P.J. & Rabinowitz, P. (1984). Methods of Numerical Integration, 2nd Ed. Academic Press Inc, Orlando FL.zbMATHGoogle Scholar
  5. Efron, B. & Stein, C. (1981). The Jackknife Estimate of Variance. Ann. Stat. 9, pp 586 – 596.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Efron, B. & Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, London.zbMATHGoogle Scholar
  7. Faure, H. (1982). Discrépance de Suites Associées à un système de Numération (en Dimension s). Acta Arith. 41, 337 – 351.MathSciNetzbMATHGoogle Scholar
  8. Genz, A. (1984). Testing Multidimensional Integration Routines, in Tools, Methods and Languages for Scientific and Engineering Computation, B. Ford, J. C. Rault, & F. Thomasset, eds., 81 – 94 North-Holland, Amsterdam.Google Scholar
  9. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.Google Scholar
  10. Joe, S. (1990). Randomization of Lattice Rules for Numerical Multiple Integration. J. Comp. Appl. Math. 31, 299 – 304.MathSciNetzbMATHCrossRefGoogle Scholar
  11. McKay, M.D., Conover, W.J. & Beckman, R.J. (1979). A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code. Technometrics 21, 239 – 245.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Niederreiter, H. (1978). Quasi-Monte Carlo Methods and Pseudo-Random Numbers. Bull. Amer. Math. Soc. 84957 – 1041.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Niederreiter, H. (1987). Point Sets and Sequences with Small Discrepancy. Monatsh. Math. 104, 273 – 337.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Niederreiter, H. (1988). Low Discrepancy and Low-Dispersion Sequences. J. Number Theory 30, 51 – 70.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF, SIAM, Philadelphia.zbMATHGoogle Scholar
  16. Owen, A. B. (1990). Empirical Likelihood and Small Samples. Computing Science and Statistics, Proceedings of the 22nd Symposium on the Interface A.S.A., Alexandria VA.Google Scholar
  17. Owen, A. B. (1992a). A Central Limit Theorem for Latin Hypercube Sampling. Journal of the Royal Statistical Society, Ser. B. Google Scholar
  18. Owen, A.B. (1992b). Orthogonal Arrays for Computer Experiments, Integration and Visualization. Stat. Sinica 2, 439 – 452.MathSciNetzbMATHGoogle Scholar
  19. Owen, A.B. (1994a). Lattice Sampling Revisited: Monte Carlo Variance of Means Over Randomized Orthogonal Arrays. Ann. Stat. 22pp. 930 – 945.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Owen, A.B. (1994b). “Monte Carlo Variance of Scrambled Equidistribution Quadrature”. Dept. of Statistics Technical Report Number 466, Stanford University.Google Scholar
  21. Patterson, H.D. (1954). The Errors of Lattice Sampling. J.R.S.S. B 16, 140 – 149.MathSciNetzbMATHGoogle Scholar
  22. Raghavarao, D. (1971). Constructions and Combinatorial Problems in Design of Experiments. Wiley, New York.zbMATHGoogle Scholar
  23. Ripley, B.D (1987). Stochastic Simulation. J. Wiley, New York.zbMATHCrossRefGoogle Scholar
  24. Sacks, J., Welch, W.J., Mitchell, T.J. & Wynn, H.P. (1989). Design and Analysis of Computer Experiments. Statistical Science 4, 409 – 435.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Roos, P. & Arnold, L. (1963). Numerische Experimente zur Mehrdimensionalen Quad- ratur. Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 172, 271 – 286.MathSciNetzbMATHGoogle Scholar
  26. Sobol’, I.M. (1967). The Distribution of Points in a Cube and the Accurate Evaluation of Integrals (in Russian). Zh. Vychisl. Mat. i Mat. Phys. 7, 784 – 802.MathSciNetGoogle Scholar
  27. Stein, M. (1987). Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics 29, 143 – 151.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Tang, B. (1993). Orthogonal Array-Based Latin Hypercubes. J.A.S.A. 88, 1392 – 1397.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Art B. Owen
    • 1
  1. 1.Dept. of StatisticsSequoia Hall Stanford UniversityStanfordUSA

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