Super-polynomial accuracy of one dimensional randomized nets using the median of means
Let $ f $ be analytic on $[0, 1] $ with $| f^{(k)}(1/2)|\leqslant A\alpha^ kk! $ for some constants
$ A $ and $\alpha< 2$ and all $ k\geqslant 1$. We show that the median estimate of
$\mu=\int _0^ 1f (x)\,\mathrm {d} x $ under random linear scrambling with $ n= 2^ m $ points
converges at the rate $ O (n^{-c\log (n)}) $ for any $ c< 3\log (2)/\pi^ 2\approx 0.21$. We also
get a super-polynomial convergence rate for the sample median of $2 k-1$ random linearly
scrambled estimates, when $ k/m $ is bounded away from zero. When $ f $ has a $ p $'th …
$ A $ and $\alpha< 2$ and all $ k\geqslant 1$. We show that the median estimate of
$\mu=\int _0^ 1f (x)\,\mathrm {d} x $ under random linear scrambling with $ n= 2^ m $ points
converges at the rate $ O (n^{-c\log (n)}) $ for any $ c< 3\log (2)/\pi^ 2\approx 0.21$. We also
get a super-polynomial convergence rate for the sample median of $2 k-1$ random linearly
scrambled estimates, when $ k/m $ is bounded away from zero. When $ f $ has a $ p $'th …
Abstract
Let be analytic on with for some constants and and all . We show that the median estimate of under random linear scrambling with points converges at the rate for any . We also get a super-polynomial convergence rate for the sample median of random linearly scrambled estimates, when is bounded away from zero. When has a ’th derivative that satisfies a -Hölder condition then the median of means has error for any , if as . The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. References
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