###### 5.9.1 Control Variates

Variance reduction techniques reduce the standard error of a Monte Carlo estimator by making the estimator more deterministic. The method of control variates accomplishes this with a simple but informative approximation. Consider crude Monte Carlo estimator

[5.55]

for some quantity ψ = *E*[*f* (** U**)],

**~**

*U**U*((0,1)

_{n}*). Let ξ be a function from*

^{n}*to for which the mean*

^{n}[5.56]

is known. We shall refer to the random variable ξ(** U**) as a

**control variate**. Consider the random variable

*f **(

**) based upon this control variate,**

*U*[5.57]

for some constant *c*. This is an unbiased estimator for ψ because

[5.58]

[5.59]

[5.60]

[5.61]

Accordingly, we can estimate ψ with the Monte Carlo estimator

[5.62]

This will have a lower standard error than [5.55] if the standard deviation σ* of *f **(** U**) is smaller than the standard deviation σ of

*f*(

**). That will happen if ξ(**

*U***) has a high correlation ρ with the random variable**

*U**f*(

**), in which case random variables**

*U**c*ξ(

**) and**

*U**f*(

**) will tend to offset each other in [5.62]. We formalize this observation by calculating**

*U*[5.63]

[5.64]

[5.65]

where σ_{ξ} is the standard deviation of ξ(** U**). Accordingly, σ* will be smaller than σ if

[5.66]

It can be shown that σ* is minimized by setting

[5.67]

in which case, from [5.65],

[5.68]

Often, ρ and σ_{ξ} are unknown, which makes determining the optimal value [5.67] for *c* problematic. We can estimate ρ and σ_{ξ} with a separate Monte Carlo analysis. Alternatively, if ξ closely approximates *f*, *c* might simply be set equal to 1.

###### 5.9.2 Example: control variates

Suppose

[5.69]

and consider the definite integral

[5.70]

This has crude Monte Carlo estimator

[5.71]

We may reduce its standard error with the control variate

[5.72]

Here, ξ is a second-order Taylor approximation for *f*. We easily obtain the mean μ_{ξ} of ξ(** U**):

[5.73]

[5.74]

[5.75]

[5.76]

We set *c* = 1 and obtain

[5.77]

[5.78]

Our control-variate Monte Carlo estimator is

[5.79]

To compare the performance of this estimator with that of the crude estimator [5.71], we set *m* = 100 and select a realization for {*U*^{[1]}, *U*^{[2]}, … ,* U*^{ [100]}}, which is presented with a histogram in Exhibit 5.13.

*f*(

*u*^{[k]}) (left) and

*f**(

*u*^{[k]}) (right) obtained as described in the text.

We evaluate the functions *f* and *f ** at each point. Results are presented as histograms in Exhibit 5.14.

*f*(

*u*^{[k]}) (top) and

*f**(

*u*^{[k]}) (bottom) obtained as described in the text.

Sample means are 1.427 and 1.430, respectively, so estimators [5.71] and [5.75] have yielded similar estimates for ψ. However, the sample standard deviations are 0.338 and 0.122, respectively, so values *f* (*U*^{[k]}) have a substantially higher sample standard deviation than values *f **(*U*^{[k]}). We can use these sample standard deviations to estimate the standard errors of our two estimators. The sample standard deviation 0.338 is an estimate for the standard deviation σ of *f* (** U**). The sample standard deviation 0.122 is an estimate for the standard deviation σ* of

*f **(

**). Substituting these into [5.35], we estimate the standard errors of our Monte Carlo estimators as 0.0338 and 0.0122, respectively. Variance reduction using a control variate reduced standard error.**

*U*To place this variance reduction in perspective, consider how much we would need to increase the sample size *m* for crude Monte Carlo estimator [5.71] to achieve the same reduction in standard error. We set [5.35] equal to 0.0122 and substitute our sample standard deviation 0.338 for σ. We obtain

[5.80]

In this particular example, our control variate estimator [5.75] accomplishes with a sample of size 100 what the crude estimator [5.71] would accomplish with a sample of size 768.

In any application, the variance reduction achieved with a particular control variate ξ(** U**) depends critically upon the correlation ρ between ξ(

**) and**

*U**f*(

**). Suppose we employ the optimal value for**

*U**c*given by [5.67], and consider the ratio of the standard error for a crude estimator [5.55] with that of a control variate estimator [5.62]. By [5.68], this is

[5.81]

If ξ(** U**) and

*f*(

**) have a correlation of ρ = .9, standard error is reduced by a factor of 1/2.3. If the correlation is .99, that factor increases to 1/7.1. In the example we just presented, the correlation was actually .992, but we only reduced standard error by a factor of 1/2.8 because we set**

*U**c*= 1 instead of determining an optimal value.