Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands

Annals of Statistics

Volumn 39, Issue 3, 2011, Pages 1372-1398

  Dick, Josef ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Digital nets; Quasi Monte Carlo; Randomized quasi Monte Carlo

Indexed keywords

EID: 79957984911     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOS880     Document Type: Article

References (21)

1. BAHVALOV, N. S. (1959). Approximate computation of multiple integrals. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 3-18. MR0115275
2. CHRESTENSON, H. E. (1955). A class of generalized Walsh functions. Pacific J. Math. 5 17-31. MR0068659
3. DICK, J. (2007). Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45 2141-2176 (electronic). MR2346374
4. DICK, J. (2008). Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46 1519-1553. MR2391005
5. DICK, J. (2009). On quasi-Monte Carlo rules achieving higher order convergence. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (P. L'Ecuyer and A. Owen, eds.) 73-96. Springer, Berlin.
6. DICK, J. and PILLICHSHAMMER, F. (2010). Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge Univ. Press, Cambridge. MR2683394.
7. HEINRICH, S. (1994). Random approximation in numerical analysis. In Functional Analysis (Essen, 1991) (K. D. Bierstedt, A. Pietsch,W. M. Ruess and D. Vogt, eds.). Lecture Notes in Pure and Appl. Math. 150 123-171. Dekker, New York. MR1241675
8. HICKERNELL, F. J. (1996). The mean square discrepancy of randomized nets. ACM Trans. Modeling Comput. Simul. 6 274-296.
9. L'ECUYER, P. and LEMIEUX, C. (2002). Recent advances in randomized quasi-Monte Carlo methods. In Modeling Uncertainty (M. Dror, P. L'Ecuyer and F. Szidarovszki, eds.). Internat. Ser. Oper. Res. Management Sci. 46 419-474. Kluwer, Boston, MA. MR1893290
10. MATOUŠEK, J. (1998). On the L2-discrepancy for anchored boxes. J. Complexity 14 527-556. MR1659004
11. MATOUSšEK, J. (1999). Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics 18. Springer, Berlin. MR1697825
12. NIEDERREITER, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics 63. SIAM, Philadelphia, PA. MR1172997
13. NOVAK, E. (1988). Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Math. 1349. Springer, Berlin. MR0971255
14. 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 (Las Vegas, NV, 1994) (H. Niederreiter and J.-S. Shiue, eds.). Lecture Notes in Statist. 106 299-317. Springer, New York. MR1445791
15. OWEN, A. B. (1997).Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34 1884-1910. MR1472202 (Pubitemid 127463576)
16. OWEN, A. B. (1997). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541-1562. MR1463564
17. OWEN, A. B. (2003). Variance with alternative scramblings of digital nets. ACM Trans. Model. Comp. Simul. 13 363-378.
18. OWEN, A. B. (2008). Local antithetic sampling with scrambled nets. Ann. Statist. 36 2319-2343. MR2458189
19. TEZUKA, S. and FAURE, H. (2003). I -binomial scrambling of digital nets and sequences. J. Complexity 19 744-757. MR2040428 (Pubitemid 38012036)
20. WALSH, J. L. (1923). A closed set of normal orthogonal functions. Amer. J. Math. 45 5-24. MR1506485
21. YUE, R.-X. and HICKERNELL, F. J. (2002). The discrepancy and gain coefficients of scrambled digital nets. J. Complexity 18 135-151. MR1895080