On the value of function evaluation location information in Monte Carlo simulation

The point estimator used in naive Monte Carlo sampling weights all the computed function evaluations equally, and it does not take into account the precise locations at which the function evaluations are made. In this note, the authors consider one-dimensional integration problems in which the integrand is twice continuously differentiable. It is shown that if the weights are suitably modified to reflect the location information present in the sample, then the convergence rate of the Monte Carlo estimator can be dramatically improved from order n’-¹’/² to order n’-², where n is the number of function evaluations computed.