We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.
翻译:我们研究一个超过$[0,1美元]的函数的集成约等于$[0,1美元]的值,其依据是,从独立随机的美元(t,m,s)美元-净额中位数得出的2美元综合估计值中值为$1美元,基数为$2美元。蚊帐由Matousek随机线性波动和数字转换随机计算。如果美元分析超过$10,1美元,那么任何一种随机计算净额估计数的误差大于$2 ⁇ -cm%2美元(s)美元乘以美元(f)乘以美元(O)(1/\sqrt{m})美元。任何美元 <3\log(2)/\\\pi2\approx0.21美元。由于这些网的分布的中位数误差为$(n ⁇ -c\log(n)/s}美元,则任何一种随机计算净额估计数的误差大于$2美元=2美元。一个独立提取的样本中位数也达到这一比率,因此,只要美元/m%2美元是零作为美元作为美元/m=1美元进行独立的精确比较的结果。