Variance with alternative scramblings of digital nets

There have been many proposals for randomizations of digital nets. Some of those proposals greatly reduce the computational burden of random scrambling. This article compares the sampling variance under different scrambling methods. Some scrambling methods adversely affect the variance, even to the extent of deteriorating the rate at which variance converges to zero. Surprisingly, a new scramble proposed here, has the effect of improving the rate at which the variance converges to zero, but so far, only for one dimensional integrands. The mean squared L2 discrepancy is commonly used to study scrambling schemes. In this case, it does not distinguish among some scrambles with different convergence rates for the variance.

[1]  A. Winsor.  Sampling techniques. , 2000, Nursing times.

[2]  Art B. Owen,et al.  Latin supercube sampling for very high-dimensional simulations , 1998, TOMC.

[3]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[4]  Alexander Keller,et al.  Fast Generation of Randomized Low-Discrepancy Point Sets , 2002 .

[5]  B. Fox.  Strategies for Quasi-Monte Carlo , 1999, International Series in Operations Research & Management Science.

[6]  Henri Faure,et al.  Variations on (0, s)-Sequences , 2001, J. Complex..

[7]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[8]  Jirí Matousek,et al.  On the L2-Discrepancy for Anchored Boxes , 1998, J. Complex..

[9]  A. Owen.  Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences , 1995 .

[10]  Art B. Owen,et al.  Monte Carlo, Quasi-Monte Carlo, and Randomized Quasi-Monte Carlo , 2000 .

[11]  P. Gruber,et al.  Funktionen von beschränkter Variation in der Theorie der Gleichverteilung , 1990 .

[12]  Fred J. Hickernell,et al.  Algorithm 823: Implementing scrambled digital sequences , 2003, TOMS.

[13]  Fred J. Hickernell,et al.  The Mean Square Discrepancy of Scrambled (t, s)-Sequences , 2000, SIAM J. Numer. Anal..

[14]  A. Owen.  Scrambled net variance for integrals of smooth functions , 1997 .

[15]  S. Haber.  A modified Monte-Carlo quadrature. II. , 1966 .

[16]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .

[17]  H. Faure.  Good permutations for extreme discrepancy , 1992 .

[18]  K. F. Roth,et al.  On irregularities of distribution IV , 1979 .

[19]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[20]  S. C. Zaremba.  Some applications of multidimensional integration by parts , 1968 .

[21]  S. Heinrich.  Random Approximation in Numerical Analysis , 1994 .

[22]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[23]  Fred J. Hickernell,et al.  The mean square discrepancy of randomized nets , 1996, TOMC.

[24]  A. Owen.  Monte Carlo Variance of Scrambled Net Quadrature , 1997 .

[25]  Shu Tezuka,et al.  I-binomial scrambling of digital nets and sequences , 2003, J. Complex..

[26]  Shu Tezuka,et al.  Another Random Scrambling of Digital ( t , s )-Sequences , 2002 .

[27]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[28]  E. Hlawka.  Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung , 1961 .

[29]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

[30]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[31]  Ken Seng Tan,et al.  Applications of randomized low discrepancy sequences to the valuation of complex securities , 2000 .

[32]  F. J. Hickernell.  Quadrature Error Bounds with Applications to Lattice Rules , 1997 .

[33]  Fred J. Hickernell,et al.  The asymptotic efficiency of randomized nets for quadrature , 1999, Math. Comput..

[34]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.