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Scrambling non-uniform nets. (English)
Math. Slovaca 59, No. 3, 379-386 (2009).
The Monte Carlo method of integration yields the probabilistic error term $\left \vert \frac {1}{N}\sum _{n=1}^Nf(\bold x_n)- \int _{[0,1]^s}f(\bold x)\,d\bold x \right \vert =O(N^{-1/2}).$ The quasi-Monte Carlo method gives the deterministic error term $O\bigl ((\log N)^s/N\bigr)$ for the best low discrepancy sequence $\bold x_n$, but for big dimension $s$ this error term is inapplicable. {\it {A. B. Owen}} [Ann. Stat. 25, No.~4, 1541‒1562 (1997; Zbl 0886.65018)] suggested to combine a deterministic sequence with random permutations of its digits, which is called scrambling. He applied it to a $(t,m,s)$-net sequence and achieved a variance of the error term of $O(N^{-3})(\log N)^{s-1}$. This paper applies Owen’s random scrambling to a special $(m-1,m,s)$-net which consists of $s$ copies of a $(0,m,1)$-net and which has a bad discrepancy, but for $L^2$ functions $f$ the author finds conditions that the variance of the integration error term is better than in a classical Monte Carlo method. Thus, Owen’s scrambling works well not only for a low-discrepancy but also for a bad discrepancy sequence. Finally, the author discusses his scrambled $(m-1, m, s)$-net from the viewpoint of the Latin hypercube sampling defined by {\it {M. D. McKay}}, {\it {R. J. Beckman}} and {\it {W. J. Conover}} [Technometrics 21, 239‒245 (1979; Zbl 0415.62011)].
Reviewer: Oto Strauch (Bratislava)