Unbiased Monte Carlo Integration Methods with Exactness for Low Order Polynomials

We consider methods for estimation of the integral of a given function that combine the unbiasedness of Monte Carlo integration (which permits a simple statistical assessment of the error) with the higher precision often attained by deterministic methods. We propose symmetric random designs with $2k + 1$ points that achieve exactness for polynomials of degree up to $2k + 1$. The distribution for the three-point method is unique, although for higher order methods there are multiple choices for the sampling distribution. For two-dimensional multiple integration over a rectangle, we propose an unbiased five-point method that achieves exactness for polynomials of degree up to three in both variables. Some bounds on error variances are given.