The Annals of Statistics

Scrambled net variance for integrals of smooth functions

Art B. Owen

Abstract

Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. One version, randomized $(t, m, s)$-nets, has the property that the integral estimates are unbiased and that the variance is $o(1/n)$, for any square integrable integrand.

Stronger assumptions on the integrand allow one to find rates of convergence. This paper shows that for smooth integrands over s dimensions, the variance is of order $n^{-3}(\log n)^{s-1}$, compared to $n^{-1}$ for ordinary Monte Carlo. Thus the integration errors are of order $n^{-3/2}(\log n)^{(s-1)/2} in probability. This compares favorably with the rate$n^{-1}(\log n)^{s-1}$for unrandomized$(t, m, s)\$-nets.

Article information

Source
Ann. Statist. Volume 25, Number 4 (1997), 1541-1562.

Dates
First available in Project Euclid: 9 September 2002

http://projecteuclid.org/euclid.aos/1031594731

Digital Object Identifier
doi:10.1214/aos/1031594731

Mathematical Reviews number (MathSciNet)
MR1463564

Subjects
Primary: 65C05: Monte Carlo methods 68O20
Secondary: 62D05: Sampling theory, sample surveys 62K05: Optimal designs

Citation

Owen, Art B. Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 (1997), no. 4, 1541--1562. doi:10.1214/aos/1031594731. http://projecteuclid.org/euclid.aos/1031594731.

References

• DAUBECHIES, I. 1992. Ten Lectures on Wavelets. SIAM, Philadelphia. Z.
• EFRON, B. and STEIN, C. 1981. The jackknife estimate of variance. Ann. Statist. 9 586 596. Z.
• GENZ, A. 1994. Testing multidimensional integration routines. In Tools, Methods and Z Languages for Scientific and Engineering Computation B. Ford, J. C. Rault and F.. Thomasset, eds. 81 94. North Holland, Amsterdam. Z.
• HICKERNELL, F. J. 1996a. Quadrature error bounds and figures of merit for quasi-random points. Technical Report 111, Dept. Mathematics, Hong Kong Baptist Univ. Z.
• HICKERNELL, F. J. 1996b. The mean square discrepancy of randomized nets. Technical Report 112, Dept. Mathematics, Hong Kong Baptist Univ. Z.
• JAWERTH, B. and SWELDENS, W. 1994. An overview of wavelet based multiresolution analyses. SIAM Review 30 377 412. Z. 2 Z n.
• MADy CH, W. R. 1992. Some elementary properties of multiresolution analyses of L R. In Z. Wavelets: A Tutorial in Theory and Applications C. K. Chui, ed. 259 294. Academic Press, New York. Z.
• MOROKOFF, W. J. and CAFLISCH, R. E. 1994. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15 1251 1279. Z.
• NIEDERREITER, H. 1992. Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia. Z.
• NIEDERREITER, H. and XING, C. 1995. Low discrepancy sequences obtained from global function fields. In Finite Fields and Applications 241 273. Cambridge Univ. Press. Z.
• OWEN, A. B. 1992. Orthogonal array s for computer experiments, integration and visualization. Statist. Sinica 2 439 452. Z. Z. Z.
• OWEN, A. B. 1995. Randomly permuted t, m, s -nets and t, s -sequences. In Monte Carlo and Z Quasi-Monte Carlo Methods in Scientific Computing H. Niederreiter and J.-S. Shiue,. eds. 299 317. Springer, New York. Z.
• OWEN, A. B. 1997. Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34. To appear. Z.
• PASKOV, S. H. 1993. Average case complexity of multivariate integration for smooth functions. J. Complexity 9 291 312. Z.
• RITTER, K. 1995. Average case analysis of numerical problems. Ph.D. dissertation, Univ. Erlangen, Germany. Z.
• RITTER, K., WASILKOWSKI, G. W. and WOZNIAKOWSKI, H. 1993. On multivariate integration for ´ Z. stochastic processes. In Numerical Integration H. Brass and G. Hammerlin, eds. 112 ¨ 331 347. Birkhauser, Basel. ¨ Z.
• WAHBA, G. 1990. Spline Models for Observational Data. SIAM, Philadelphia. Z.
• WASILKOWSKI, G. W. 1993. Integration and approximation of multivariate functions: average case complexity with isotropic Wiener measure. Bull. Amer. Math. Soc. 28 308 314.
• WOZNIAKOWSKI, H. 1991. Average case complexity of multivariate integration. Bull. Amer. ´ Math. Soc. 24 185 194.
• STANFORD, CALIFORNIA 94305 E-MAIL: owen@play fair.stanford.edu