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

Let f be analytic on [0, 1] with |f (1/2)| 6 Aαk! for some constant A and α < 2. We show that the median estimate of μ = ∫ 1 0 f(x) dx under random linear scrambling with n = 2 points converges at the rate O(n−c ) for any c < 3 log(2)/π ≈ 0.21. We also get a superpolynomial 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)+ ) for any > 0, if k →∞ as m→∞.