The Annals of Statistics

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

Josef Dick

Full-text: Open access

Abstract

We study a random sampling technique to approximate integrals [0, 1]sf(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics.

The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the number of samples) for functions f with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order N−3/2+ε under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order N−3/2−1/s+ε (where s ≥ 1 is the dimension) for functions with mixed partial derivatives up to order two.

Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate.

This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order Nα−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 1 in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order N−5/2 and N−7/2, in accordance with the theoretical upper bound.

Article information

Source
Ann. Statist. Volume 39, Number 3 (2011), 1372-1398.

Dates
First available in Project Euclid: 4 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1304514657

Digital Object Identifier
doi:10.1214/11-AOS880

Mathematical Reviews number (MathSciNet)
MR2850206

Zentralblatt MATH identifier
1298.65011

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 65D32: Quadrature and cubature formulas

Keywords
Digital nets randomized quasi-Monte Carlo quasi-Monte Carlo

Citation

Dick, Josef. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39 (2011), no. 3, 1372--1398. doi:10.1214/11-AOS880. https://projecteuclid.org/euclid.aos/1304514657.


Export citation

References

  • [1] Bahvalov, N. S. (1959). Approximate computation of multiple integrals. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 3–18.
  • [2] Chrestenson, H. E. (1955). A class of generalized Walsh functions. Pacific J. Math. 5 17–31.
  • [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).
  • [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.
  • [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.
  • [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.
  • [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.
  • [10] Matoušek, J. (1998). On the L2-discrepancy for anchored boxes. J. Complexity 14 527–556.
  • [11] Matoušek, J. (1999). Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics 18. Springer, Berlin.
  • [12] Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics 63. SIAM, Philadelphia, PA.
  • [13] Novak, E. (1988). Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Math. 1349. Springer, Berlin.
  • [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.
  • [15] Owen, A. B. (1997). Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34 1884–1910.
  • [16] Owen, A. B. (1997). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541–1562.
  • [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.
  • [19] Tezuka, S. and Faure, H. (2003). I-binomial scrambling of digital nets and sequences. J. Complexity 19 744–757.
  • [20] Walsh, J. L. (1923). A closed set of normal orthogonal functions. Amer. J. Math. 45 5–24.
  • [21] Yue, R.-X. and Hickernell, F. J. (2002). The discrepancy and gain coefficients of scrambled digital nets. J. Complexity 18 135–151.