Local antithetic sampling with scrambled nets
Art B. Owen
Source: Ann. Statist. Volume 36, Number 5
(2008), 2319-2343.
Abstract
We consider the problem of computing an approximation to the integral I=∫[0, 1]d f(x) dx. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of O(n−1/2) from n independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate O(n−1+ɛ) for any ɛ>0 and randomized QMC (RQMC) can attain the RMSE O(n−3/2+ɛ), both under mild conditions on f.
Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining O(n−3/2−1/d+ɛ) for any ɛ>0, for smooth enough f.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1223908094
Digital Object Identifier: doi:10.1214/07-AOS548
Mathematical Reviews number (MathSciNet): MR2458189
Zentralblatt MATH identifier: 1157.65006
References
[1] Acworth, P., Broadie, M. and Glasserman, P. (1997). A comparison of some Monte Carlo techniques for option pricing. In Monte Carlo and Quasi-Monte Carlo Methods’96 (H. Niederreiter, P. Hellekalek, G. Larcher and P. Zinterhof, eds.) 1–18. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1644509
Zentralblatt MATH: 0888.90010
[2] Caflisch, R., Morokoff, E. W. and Owen, A. B. (1997). Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Computational Finance 1 27–46.
[3] Chelson, P. (1976). Quasi-random techniques for Monte Carlo methods. Ph.D. thesis, The Claremont Graduate School.
[4] Cools, R. (1999). Monomial cubature rules since Stroud: A compilation—part 2. J. Comput. Appl. Math. 112 21–27.
Mathematical Reviews (MathSciNet): MR1728449
Zentralblatt MATH: 0954.65021
Digital Object Identifier: doi:10.1016/S0377-0427(99)00229-0
[5] Cools, R. and Rabinowitz, P. (1993). Monomial cubature rules since Stroud: A compilation. J. Comput. Appl. Math. 48 309–326.
Mathematical Reviews (MathSciNet): MR1252544
Digital Object Identifier: doi:10.1016/0377-0427(93)90027-9
[6] Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596.
Mathematical Reviews (MathSciNet): MR615434
Zentralblatt MATH: 0481.62035
Digital Object Identifier: doi:10.1214/aos/1176345462
Project Euclid: euclid.aos/1176345462
[7] Faure, H. (1982). Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41 337–351.
[8] Fishman, G. S. (2006). A First Course in Monte Carlo. Duxbury, Belmont, CA.
[9] Fox, B. L. (1999). Strategies for Quasi-Monte Carlo. Kluwer Academic, Boston.
[10] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.
Mathematical Reviews (MathSciNet): MR1999614
Zentralblatt MATH: 1038.91045
[11] Haber, S. (1970). Numerical evaluation of multiple integrals. SIAM Rev. 12 481–526.
Mathematical Reviews (MathSciNet): MR285119
Zentralblatt MATH: 0206.46905
Digital Object Identifier: doi:10.1137/1012102
JSTOR: links.jstor.org
[12] Hickernell, F. J., Lemieux, C. and Owen, A. B. (2005). Control variates for quasi-Monte Carlo (with discussion). Statist. Sci. 20 1–31.
Mathematical Reviews (MathSciNet): MR2182985
Digital Object Identifier: doi:10.1214/088342304000000468
Project Euclid: euclid.ss/1118065040
[13] Hlawka, E. (1961). Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl. 54 325–333.
Mathematical Reviews (MathSciNet): MR139597
Digital Object Identifier: doi:10.1007/BF02415361
[14] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
Mathematical Reviews (MathSciNet): MR26294
Zentralblatt MATH: 0032.04101
Digital Object Identifier: doi:10.1214/aoms/1177730196
Project Euclid: euclid.aoms/1177730196
[15] L’Ecuyer, P. and Lemieux, C. (2002). A survey of 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 Academic, New York.
[16] Matoušek, J. (1998). Geometric Discrepancy: An Illustrated Guide. Springer, Heidelberg.
[17] Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet): MR1172997
Zentralblatt MATH: 0761.65002
[18] Niederreiter, H. and Pirsic, G. (2001). The microstructure of (t, m, s)-nets. J. Complexity 17 683–696.
Mathematical Reviews (MathSciNet): MR1881664
Zentralblatt MATH: 0997.11059
Digital Object Identifier: doi:10.1006/jcom.2001.0596
[19] Owen, A. B. (1995). Randomly permuted (t, m, s)-nets and (t, s)-sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.) 299–317. Springer, New York.
Mathematical Reviews (MathSciNet): MR1445777
[20] Owen, A. B. (1997). Monte Carlo variance of scrambled equidistribution quadrature. SIAM J. Numer. Anal. 34 1884–1910.
Mathematical Reviews (MathSciNet): MR1472202
Zentralblatt MATH: 0890.65023
Digital Object Identifier: doi:10.1137/S0036142994277468
JSTOR: links.jstor.org
[21] Owen, A. B. (1997). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541–1562.
Mathematical Reviews (MathSciNet): MR1463564
Digital Object Identifier: doi:10.1214/aos/1031594731
Project Euclid: euclid.aos/1031594731
[22] Owen, A. B. (1998). Scrambling Sobol’ and Niederreiter–Xing points. J. Complexity 14 466–489.
Mathematical Reviews (MathSciNet): MR1659008
Digital Object Identifier: doi:10.1006/jcom.1998.0487
[23] Owen, A. B. (2003). Variance with alternative scramblings of digital nets. ACM Trans. Modeling and Computer Simulation 13 363–378.
[24] Schürer, R. and Schmid, W. C. (2006). MinT: A database for optimal net parameters. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (H. Niederreiter and D. Talay, eds.) 457–469. Springer, Berlin.
[25] Sloan, I. H. and Joe, S. (1994). Lattice Methods for Multiple Integration. Oxford Science Publications.
Mathematical Reviews (MathSciNet): MR1442955
[26] Sobol’, I. M. (1967). The use of Haar series in estimating the error in the computation of infinite-dimensional integrals. Dokl. Akad. Nauk SSSR 8 810–813.
Mathematical Reviews (MathSciNet): MR215527
[27] Spanier, J. and Maize, E. H. (1994). Quasi-random methods for estimating integrals using relatively small samples. SIAM Rev. 36 18–44.
Mathematical Reviews (MathSciNet): MR1267048
Digital Object Identifier: doi:10.1137/1036002
JSTOR: links.jstor.org
[28] Stroud, A. H. (1971). Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, NJ.
Mathematical Reviews (MathSciNet): MR327006
Zentralblatt MATH: 0379.65013
[29] Takemura, A. (1983). Tensor analysis of ANOVA decomposition. J. Amer. Statist. Assoc. 78 894–900.
Mathematical Reviews (MathSciNet): MR727575
Zentralblatt MATH: 0535.62061
Digital Object Identifier: doi:10.2307/2288201
JSTOR: links.jstor.org
[30] Tezuka, S. and Faure, H. (2003). I-binomial scrambling of digital nets and sequences. J. Complexity 19 744–757.
Mathematical Reviews (MathSciNet): MR2040428
Zentralblatt MATH: 1073.68033
Digital Object Identifier: doi:10.1016/S0885-064X(03)00035-9
[31] Yue, R. X. and Hickernell, F. J. (2002). The discrepancy and gain coefficients of scrambled digital nets. J. Complexity 18 135–151.
Mathematical Reviews (MathSciNet): MR1895080
Zentralblatt MATH: 1114.11306
Digital Object Identifier: doi:10.1006/jcom.2001.0630
