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 onfor any c < 3 log(2)/π 2 ≈ 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 → ∞.We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of 2r − 1 independent draws.

Importance Sampling methods allow to substantially reduce the number of trials in estimation of the rare failure probability or other stochastic metrics. These methods can be viewed as a rigorous generalization of quantitative "torture" or "stress" methods where the process is artificially modified to increase the probability of failure, and the failure probability estimations obtained for such modified process are extrapolated to the original process with rare failures. Applications of Importance Sampling methods are presented and demonstrated on computationally efficient estimations of via failure probability and via LCDU. The accuracy of the Importance Sampling LCDU estimates is verified by comparison with experimental results. Applications of Importance Sampling methods to experimental measurements are discussed.

Let f be analytic on [0, 1] with |f (k) (1/2)| Aα k k! 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 m points converges at the rate O(n −c log(n) ) for any c < 3 log(2)/π 2 ≈ 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 → ∞.

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