Control variates for quasi-Monte Carlo

Statistical Science

Volumn 20, Issue 1, 2005, Pages 1-31

  Hickernell, Fred J ^a   Lemieux, Christiane ^b   Owen, Art B ^c   L'Ecuyer, Pierre ^d   Meng, Xiao Li ^e  

^a Illinois Institute of Technology   (United States)

^b UNIVERSITY OF CALGARY   (Canada)

^c STANFORD UNIVERSITY   (United States)

^d UNIVERSITÉ DE MONTRÉAL   (Canada)

^e HARVARD UNIVERSITY   (United States)

Author keywords

Digital nets; Lattice rules; Low discrepancy methods; Stratification; Variance reduction

Indexed keywords

EID: 22544472182     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/088342304000000468     Document Type: Article

References (90)

1. AVRAMIDIS, A. and WILSON, J. R. (1993). A splitting scheme for control variates. Oper. Res. Lett. 14 187-198.
2. BECK, J. and CHEN, W. W. L. (1987). Irregularities of Distribution. Cambridge Univ. Press.
3. BEN AMEUR, H., L'ECUYR, P. and LEMIEUX, C. (1999). Variance reduction of Monte Carlo and randomized quasi-Monte Carlo estimators for stochastic volatility models in finance. In Proc. 1999 Winter Simulation Conference 1 336-343. IEEE Press, New York.
4. BRATLEY, P., Fox, B. L. and SCHRAGE, L. E. (1987). A Guide to Simulation, 2nd ed. Springer, New York.
5. CAFLISCH, R. E., MOROKOFF, W. and OWEN, A. B. (1997). Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension. J. Comput. Finance 1 27-46.
6. CHELSON, P. (1976). Quasi-random techniques for Monte Carlo methods. Ph.D. dissertation, Claremont Graduate School.
7. COCHRAN, W. G. (1977). Sampling Techniques, 3rd ed. Wiley, New York.
8. CRANLEY, R. and PATTERSON, T. (1976). Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13 904-914.
9. EFRON, B. and STEIN, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586-596.
10. FANG, K.-T. and WANG, Y. (1994). Number-Theoretic Methods in Statistics. Chapman and Hall, London.
11. FAURE, H. (1982). Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41 337-351.
12. FISHMAN, G. (1996). Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York.
13. HEINRICH, S., HICKERNELL, F. J. and YUE, R.-X. (2004). Optimal quadrature for Haar wavelet spaces. Math. Comp. 73 259-277.
14. HICKERNELL, F. J. (1996). Quadrature error bounds with applications to lattice rules. SIAM J. Numer. Anal. 33 1995-2016; corrected printing of Sections 3-6 (1997) 34 853-866.
15. HICKERNELL, F. J., HONG, H. S., L'ECUYER, P. and LEMIEUX, C. (2000). Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22 1117-1138.
16. HICKERNELL, F. J. and NIEDERREITER, H. (2003). The existence of good extensible rank-1 lattices. J. Complexity 19 286-300.
17. HICKERNELL, F. J. and YUE, R.-X. (2000). The mean square discrepancy of scrambled (t, s)-sequences. SIAM J. Numer. Anal. 38 1089-1112.
18. HOEFFDING, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293-325.
19. HONG, H. S. and HICKERNELL, F. J. (2003). Algorithm 823: Implementing scrambled digital sequences. AMS Trans. Math. Software 29 95-109.
20. HUA, L. and WANG, Y. (1981). Applications of Number Theory to Numerical Analysis. Springer, Berlin.
21. KOROBOV, N. M. (1959). The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR 124 1207-1210.
22. KUIPERS, L. and NIEDERREITER, H. (1974). Uniform Distribution of Sequences. Wiley, New York.
23. L'ECUYER, P. and LEMIEUX, C. (2002). Recent advances in randomized quasi-Monte Carlo methods. In Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications (M. Dror, P. L'Ecuyer and F. Szidarovszki, eds.) 419-474. Kluwer, Dordrecht.
24. LIAO, J. G. (1998). Variance reduction in Gibbs sampler using quasi random numbers. J. Comput. Graph. Statist. 7 253-266.
25. LOH, W.-L. (2003). On the asymptotic distribution of scrambled net quadrature. Ann. Statist. 31 1282-1324.
26. LOHR, S. (1999). Sampling: Design and Analysis. Brooks/Cole., Pacific Grove, CA.
27. 2 -discrepancy for anchored boxes. J. Complexity 14 527-556.
28. MATOUŠEK, J. (1999). Geometric Discrepancy. An Illustrated Guide. Springer, Heidelberg.
29. MOROKOFF, W. and CAFLISCH, R. E. (1995). Quasi-Monte Carlo integration. J. Comput. Phys. 122 218-230.
30. NIEDERREITER, H. (1987). Point sets and sequences with small discrepancy. Monatsh. Math. 104 273-337.
31. NIEDERREITER, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia.
32. NIEDERREITER, H. and PIRSIC, G. (2001). The microstructure of (t, m, s)-nets. J. Complexity 17 683-696.
33. NIEDERREITER, H. and XING, C. (2001). Rational Points on Curves over Finite Fields: Theory and Applications. London Math. Soc. Lecture Note Ser. 285. Cambridge Univ. Press.
34. OSTLAND, M. and Yu, B. (1997). Exploring quasi Monte Carlo for marginal density approximation. Statist. Comput. 7 217-228.
35. OWEN, A. B. (1992). A central limit theorem for Latin hypercube sampling. J. Roy. Statist. Soc. Ser. B 54 541-551.
36. OWEN, A. B. (1995). Randomly permuted (t, m, s)-nets and (t, s)-sequences. Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statist. 106 299-317. Springer, New York.
37. OWEN, A. B. (1997a). Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34 1884-1910.
38. OWEN, A. B. (1997b). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541-1562.
39. OWEN, A. B. (1998a). Latin supercube sampling for very high dimensional simulations. ACM Transactions on Modeling and Computer Simulation 8 71-102.
40. OWEN, A. B. (1998b). Scrambling Sobol' and Niederreiter-Xing points. J. Complexity 14 466-489.
41. OWEN, A. B. (2002). Scrambled net variance with alternative scramblings. Technical report, Dept. Statistics, Stanford Univ.
42. PASKOV, S. (1993). Average case complexity of multivariate integration for smooth functions. J. Complexity 9 291-312.
43. RIPLEY, B.D. (1987). Stochastic Simulation. Wiley, New York.
44. RITCHKEN, P., SANKARASUBRAMANIAN, L. and VIJH, A. M. (1993). The valuation of path dependent contracts on the average. Management Sci. 39 1202-1213.
45. SARKAR, P. K. and PRASAD, M. A. (1987). A comparative study of pseudo- and quasirandom sequences for the solution of integral equations. J. Comput. Phys. 68 66-88.
46. SCHLIER, C. (2002). A practitioner's view on QMC integration. Technical report, Fakultät für Physik, Univ. Freiburg.
47. SLOAN, I. H. and JOE, S. (1994). Lattice Methods for Multiple Integration, Oxford Univ. Press, New York.
48. SLOAN, I. H., Kuo, F. Y. and JOE, S. (2002a). Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40 1650-1665.
49. SLOAN, I. H., KUO, F. Y. and JOE, S. (2002b). On the step-by-step construction of quasi-Monte Carlo integation rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comp. 71 1609-1640.
50. SLOAN, I. H. and REZTSOV, A. V. (2002). Component-by-component construction of good lattice rules. Math. Comp. 71 263-273.
51. SOBOL', I. M. (1967). The distribution of points in a cube and the accurate evaluation of integrals. Zh. Vychisl. Mat. i Mat. Fiz. 7 784-802. (In Russian.)
52. SOBOL', I. M. (1969). Multidimensional Quadrature Formulas and Haar Functions. Nauka, Moscow. (In Russian.)
53. SPANIER, J. and MAIZE, E. H. (1994). Quasi-random methods for estimating integrals using relatively small samples. SIAM Rev. 36 18-44.
54. TEZUKA, S. (1995). Uniform Random Numbers: Theory and Practice. Kluwer, Boston.
55. VAN DER CORPUT, J. G. (1935a). Verteilungsfunktionen I. Nederlandse Akademie van Wetenschappen Proceedings 38 813-821.
56. VAN DER CORPUT, J. G. (1935b). Verteilungsfunktionen II. Nederlandse Akademie van Wetenschappen Proceedings 38 1058-1066.
57. WEYL, H. (1914). Über ein Problem aus dem Gebeite der diophantischen Approximationen. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 234-244.
58. WEYL, H. (1916). Über die Gleichverteilung von Zahlen mod. eins. Math. Ann. 77 313-352.
59. YUE, R.-X. (1999). Variance of quadrature over scrambled unions of nets. Statist. Sinica 9 451-473.
60. YUE, R.-X. and HICKERNELL, F. J. (2002). The discrepancy and gain coefficients of scrambled digital nets. J. Complexity 18 135-151.
61. ZAREMBA, S. K. (1968). Some applications of multidimensional integration by parts. Ann. Polon. Math. 21 85-96.
62. AVRAMIDIS, A. N., DESLAURIERS, A. and L'ECUYER, P. (2004). Modeling daily arrivals to a telephone call center. Management Sci. 50 896-908.
63. AVRAMIDIS, A. N. and WILSON, J. R. (1996). Integrated variance reduction strategies for simulation. Oper. Res. 44 327-346.
64. CASELLA, G. (1996). Statistical inference and Monte Carlo algorithms (with discussion). Test 5 249-344.
65. CASELLA, G., LAVINE, M. and ROBERT, C. P. (2001). Explaining the perfect sampling. Amer. Statist. 55 299-305.
66. CRAIU, R. and MENG, X.-L. (2005). Multiprocess parallel antithetic coupling for backward and forward Markov chain Monte Carlo. Ann. Statist. 33 661-697.
67. DIACONIS, P. (1988). Bayesian numerical analysis. In Statistical Decision Theory and Related Topics IV (S. Gupta and J. O. Berger, eds.) 1 163-175. Springer, Berlin.
68. FRIGESSI, A., GÅSEMYR, J. and RUE, H. (2000). Antithetic coupling of two Gibbs sampler chains. Ann. Statist. 28 1128-1149.
69. GELMAN, A. and MENG, X.-L. (1998). Computing normalizing constants: From importance sampling to bridge sampling to path sampling. Statist. Sci. 13 163-185.
70. GELMAN, A. and RUBIN, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci. 7457-511.
71. GLYNN, P. W. (1994). Efficiency improvement techniques. Ann. Oper. Res. 53 175-197.
72. GLYNN, P. W. and SZECHTMAN, R. (2002). Some new perspectives on the method of control variates. In Monte Carlo and Quasi-Monte Carlo Methods 2000 (K.-T. Fang, F. J. Hickernell and H. Niederreiter, eds.) 27-49. Springer, Berlin.
73. HAMMERSLEY, J. M. and MORTON, K. V. (1956). A new Monte Carlo technique: Antithetic variates. Proc. Cambridge Philos. Soc. 52 449-475.
74. HICKERNELL, F. J. (2000). What affects the accuracy of quasi-Monte Carlo quadrature? In Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.) 16-55. Springer, Berlin.
75. JIANG, T. (2003). Data driven shrinkage strategies for quasi-regression. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
76. JIANG, T. and OWEN, A. B. (2003). Quasi-regression with shrinkage. Math. Comput. Simulation 62 231-241.
77. KONG, A., MCCULLAGH, P., MENG, X.-L., NICOLAE, D. and TAN, Z. (2003). A theory of statistical models for Monte Carlo integration (with discussion). J. R. Stat. Soc. Ser B Stat. Methodol. 65 585-618.
78. Liu, C., RUBIN, D. B. and Wu, Y. (1998). Parameter expansion to accelerate EM: The PX-EM algorithm. Biometrika 85 755-770.
79. MENG, X.-L. and SCHILLING, S. (2002). Warp bridge sampling. J. Comput. Graph. Statist. 11 552-586.
80. MENG, X.-L. and WONG, W. (1996). Simulating ratios of normalizing constants via a simple identity: A theoretical exploration. Statist. Sinica 6 831-860.
81. OGATA, Y. (1989). A Monte Carlo method for high-dimensional integration. Numer. Math. 55 137-157.
82. OWEN, A. B. (2001). Empirical Likelihood. Chapman and Hall/CRC Press, Boca Raton, FL.
83. PICHITLAMKEN, J., DESLAURIERS, A., L'ECUYER, P. and AVRAMIDIS, A. N. (2003). Modeling and simulation of a telephone call center. In Proc. 2003 Winter Simulation Conference 2 1805-1812. IEEE Press, New York.
84. RITTER, K. (2000). Average-Case Analysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, Berlin.
85. SACKS, J. and YLVISAKER, D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37 66-89.
86. SACKS, J. and YLVISAKER, D. (1968). Designs for regression problems with correlated errors; many parameters. Ann. Math. Statist. 39 49-69.
87. SACKS, J. and YLVISAKER, D. (1970a). Design for regression problems with correlated errors. III. Ann. Math. Statist. 41 2057-2074.
88. SACKS, J. and YLVISAKER, D. (1970b). Statistical designs and integral approximation. In Proc. Twelfth Biennial Seminar of the Canadian Mathematics Congress (R. Pyke, ed.) 115-136. Canadian Math. Soc., Montreal.
89. TAN, Z. (2003). A likelihood approach for Monte Carlo integration. Ph.D. dissertation, Dept. Statistics, Univ. Chicago.
90. WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.