## 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, we 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 |