Higher-order Monte Carlo through cubic stratification

We propose two novel unbiased estimators of the integral\int [0,1]sf(u)du for a function f, which depend on a smoothness parameter r\in \Bbb N. The first estimator integrates exactly the polynomials of degrees p < rand achieves the optimal err or n - 1/2 - r/s(where n is the number of evaluations off) when f is rimes continuously differentiable. The second estimator is also optimal in terms of convergence rate and has the advantage of being 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 variate s based on numerical derivatives.We provide numerical evidence that they show good performance even for moderate values of n.