Super-polynomial accuracy of one dimensional randomized nets using the median-of-means

11/24/2021

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by   Zexin Pan, et al.

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Let f be analytic on [0,1] with |f^(k)(1/2)|≤ Aα^kk! for some constant A and α<2. We show that the median estimate of μ=∫_0^1f(x) dx under random linear scrambling with n=2^m points converges at the rate O(n^-clog(n)) for any c< 3log(2)/π^2≈ 0.21. We also get a super-polynomial convergence rate for the sample median of 2k-1 random linearly scrambled estimates, when k=Ω(m). When f has a p'th derivative that satisfies a λ-Hölder condition then the median-of-means has error O( n^-(p+λ)+ϵ) for any ϵ>0, if k→∞ as m→∞.