Processing simulation output by riemann sums

Abstract

When estimating an integral by a Monte-Carlo method, an alternative to the standard average of observed values is to use Riemann sums, that is trapezoidal approximations. This method has been considered in Yakowitz et al. (1977) in the special case of a uniform distribution. We generalize their approach and show that the use of Riemann sums leads to an improvement on the Monte-Carlo estimator in terms of convergence rate, since it reduces the variance by an order of magnitude, that is from l/n to l/n ². Moreover, simulations in usual settings illustrate, the surprising stability of the method.

Keywords:

• Monte Carlo methods
• order statistics
• Riemann integration