Abstract
In this paper we develop a framework to study the dependence structure of scrambled (t,m,s)-nets. It relies on values denoted by Cb(𝒌;Pn), which are related to how many distinct pairs of points from Pn lie in the same elementary 𝒌-interval in base b. These values quantify the equidistribution properties of Pn in a more informative way than the parameter t. They also play a key role in determining if a scrambled set ˜Pn is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if Cb(𝒌;Pn)≤1 for all 𝒌∈ℕs, which in turn implies that a scrambled digital (t,m,s)-net in base b is NLOD if and only if t=0. Through numerical examples we demonstrate that these Cb(𝒌;Pn) values are a powerful tool to compare the quality of different (t,m,s)-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.
Funding source: Natural Sciences and Engineering Research Council of Canada
Award Identifier / Grant number: 238959
Funding source: Austrian Science Fund
Award Identifier / Grant number: F5506-N26
Award Identifier / Grant number: F5509-N26
Funding statement: The authors wish to acknowledge the support of the Natural Science and Engineering Research Council (NSERC) of Canada for its financial support via grant #238959. The first author is also partially supported by the Austrian Science Fund (FWF): Projects F5506-N26 and F5509-N26, which are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
A Appendix
Proof of Lemma 2.5.
When either x or y is 1, from Lemma 2.4 we know that
and thus bVi-Vi-1=0 in this case. So for the remainder of the proof, we assume x,y∈[0,1). Let
be the base-b digital expansion of x and y chosen so that only finitely many digits are non-zero.
Recall that
and
Case 1:
Case 2:
because
Case 3:
Multiply by
which will be shown to be non-negative. Note that by assumption
Case 3a:
Case 3b:
Case 3c:
Case 4:
is greater than or equal to zero. Using the identities
write
and
and
Now substituting into (A.1) and simplifying, we get
By multiplying the above by
is non-negative.
Case 4a:
Case 4b:
References
[1] J. Dick, F. Y. Kuo and I. H. Sloan, High-dimensional integration: The quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288. 10.1017/S0962492913000044Search in Google Scholar
[2] J. Dick and F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. 10.1017/CBO9780511761188Search in Google Scholar
[3] H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith. 41 (1982), no. 4, 337–351. 10.4064/aa-41-4-337-351Search in Google Scholar
[4] H. Faure and C. Lemieux, Implementation of irreducible Sobol’ sequences in prime power bases, Math. Comput. Simulation 161 (2019), 13–22. 10.1016/j.matcom.2018.08.015Search in Google Scholar
[5] M. Gerber, On integration methods based on scrambled nets of arbitrary size, J. Complexity 31 (2015), no. 6, 798–816. 10.1016/j.jco.2015.06.001Search in Google Scholar
[6] M. Gnewuch, M. Wnuk and N. Hebbinghaus, On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds, Discrepancy Theory, Radon Ser. Comput. Appl. Math. 26, De Gruyter, Berlin (2020), 43–68. 10.1515/9783110652581-003Search in Google Scholar
[7] F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274–296. 10.1145/240896.240909Search in Google Scholar
[8] H. S. Hong and F. J. Hickernell, Algorithm 823: implementing scrambled digital sequences, ACM Trans. Math. Software 29 (2003), no. 2, 95–109. 10.1145/779359.779360Search in Google Scholar
[9] C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling, Springer Ser. Statist., Springer, New York, 2009. 10.1007/978-0-387-78165-5_5Search in Google Scholar
[10] C. Lemieux, Negative dependence, scrambled nets, and variance bounds, Math. Oper. Res. 43 (2018), no. 1, 228–251. 10.1287/moor.2017.0861Search in Google Scholar
[11]
J. Matoušek,
On the
[12] R. B. Nelsen, An Introduction to Copulas, 2nd ed., Springer Ser. Statist., Springer, New York, 2006. Search in Google Scholar
[13] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, Society for Industrial and Applied Mathematics, Philadelphia, 1992. 10.1137/1.9781611970081Search in Google Scholar
[14]
A. B. Owen,
Randomly permuted
[15] A. B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), no. 5, 1884–1910. 10.1137/S0036142994277468Search in Google Scholar
[16] A. B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Statist. 25 (1997), no. 4, 1541–1562. 10.1214/aos/1031594731Search in Google Scholar
[17] A. B. Owen, Scrambling Sobol’ and Niederreiter–Xing points, J. Complexity 14 (1998), no. 4, 466–489. 10.1006/jcom.1998.0487Search in Google Scholar
[18] A. B. Owen, Variance and discrepancy with alternative scramblings, ACM Trans. Model. Comput. Simul. 13 (2003), 363–378. 10.1145/945511.945518Search in Google Scholar
[19] I. M. Sobol’, On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comp. Math. Math. Phys. 7 (1967), 86–112. 10.1016/0041-5553(67)90144-9Search in Google Scholar
[20] I. M. Sobol’ and D. I. Asotsky, One more experiment on estimating high-dimensional integrals by quasi-Monte Carlo methods, Math. Comput. Simul. 62 (2003), 255–263. 10.1016/S0378-4754(02)00228-8Search in Google Scholar
[21] M. Wnuk and M. Gnewuch, Note on pairwise negative dependence of randomly shifted and jittered rank-1 lattices, Oper. Res. Lett. 48 (2020), no. 4, 410–414. 10.1016/j.orl.2020.04.005Search in Google Scholar
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