Higher-order stochastic integration through cubic stratification
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We propose two novel unbiased estimators of the integral ∫_[0,1]^sf(u) du for a function f, which depend on a smoothness parameter r∈ℕ. The first estimator integrates exactly the polynomials of degrees p<r and achieves the optimal error n^-1/2-r/s (where n is the number of evaluations of f) when f is r times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of [0,1]^s. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of n.
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