16.4 Variance reduction

The cost of precision can be cut by cleverer sampling.

• Antithetic variates: use each uniform \(U\) twice, as \(U\) and \(1 - U\), and average \(g(U)\) and \(g(1 - U)\). If \(g\) is monotone the two are negatively correlated, so the average of a pair has smaller variance than the average of two independent draws.
• Common random numbers: when comparing two policies, run both on the same random demands, arrivals or failures. The difference of the outputs then reflects the policies, not the luck of the draw, and its variance is much smaller. The \((s, S)\) search of the section on inventory used this.
• Control variates subtract a correlated quantity with known mean, and importance sampling draws more often from the important region and reweights; the latter is indispensable for estimating rare events such as system failures.

For \(\mathbb E[e^U] = e - 1\), antithetic pairs reduce the variance by a factor of \(30\) at equal cost; comparing two newsvendor orders with common demands reduces the standard error of the difference by a factor of \(5\) (Software 16.5.3).

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