In statistics, the antithetic variates method is a variance reduction technique used in Monte Carlo methods. Considering that the error reduction in the simulated signal (using Monte Carlo methods) has a square root convergence, a very large number of sample paths is required to obtain an accurate result. The antithetic variates method reduces the variance of the simulation results.
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Underlying principle
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path
Suppose that we would like to estimate
For that we have generated two samples
An unbiased estimate of
And
so variance is reduced if
Example 1
If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be
Example 2: integral calculation
We would like to estimate
The exact result is
and U follows a uniform distribution [0, 1].
The following table compares the classical Monte Carlo estimate (sample size: 2n, where n = 1500) to the antithetic variates estimate (sample size: n, completed with the transformed sample 1 − ui):
The use of the antithetic variates method to estimate the result shows an important variance reduction.