Stochastic Simulation

CHAPTER 5

Variance Reduction

Careful design of a simulation experiment can almost always improve its effectiveness for a given cost, or reduce its cost for prescribed effectiveness. That is, the cost in computer time, for it is possible for the thought in the design process to outweigh the savings (as is the case for all the examples of this chapter). This suggests that we should be looking for variance reductions of at least a factor of 2 and preferably 10 or more. Another factor to bear in mind is the ubiquitous law of statistical variation, so to reduce the standard error of an estimator by a factor of f one needs to increase the size of the experiment by around f². This means that large increases in computer power are needed to produce relative modest increases in precision. Another consequence is that it is conventional to quote variance reduction, not standard error reduction, as the cost reduction should be roughly proportional to the variance reduction.

How then can we achieve appreciable variance reductions? Many of the standard techniques are adaptations of ideas from sampling theory or the design of experiments. Both these subjects are of interest for simulation and can help suggest further dodges. Many techniques fall into one of the following categories.

(a) Importance sampling.   This involves using a distribution different from the one specified in the problem, and ...