Variance Reduction Methods I 1 Simulation Efficiency

Suppose as usual that we wish to estimate θ := E[h(X)]. Then the standard simulation algorithm is: 2. Estimate θ with θ n = n j=1 Y j /n where Y j := h(X j). 3. Approximate 100(1 − α)% confidence intervals are then given by θ n − z 1−α/2 σ n √ n , θ n + z 1−α/2 σ n √ n where σ n is the usual estimate of Var(Y) based on Y 1 ,. .. , Y n. One way to measure the quality of the estimator, θ n , is by the half-width, HW , of the confidence interval. For a fixed α, we have HW = z 1−α/2 Var(Y) n. We would like HW to be small, but sometimes this is difficult to achieve. This may be because Var(Y) is too large, or too much computational effort is required to simulate each Y j so that n is necessarily small, or some combination of the two. As a result, it is often imperative to address the issue of simulation efficiency. 1 There are a number of things we can do: 1. Develop a good simulation algorithm 2. Program carefully to minimize storage requirements. For example we do not need to store all the Y j 's: we only need to keep track of Y j and Y 2 j to compute θ n and approximate CI's 3. Program carefully to minimize execution time 4. Decrease the variability of the simulation output that we use to estimate θ. The techniques used to do this are usually called variance reduction techniques We will now study various variance reduction techniques, and assume that we are doing items (1) to (3) as well as possible. Before proceeding to study these techniques, however, we should first describe a measure of simulation efficiency. 1 Recall that as a tool, simulation is often used to solve problems that are too hard to solve either explicitly or numerically. These problems are therefore often very demanding from a computational point of view. The argument that we will not have to care about simulation efficiency as computers become ever faster is spurious: our goals will simply adapt so that we will want to solve ever harder problems.