In statistics, the antithetic variates method is a variance reduction technique used in Monte Carlo methods.
Considering that the error in the simulated signal (using Monte Carlo methods) has a one-over 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.
[1][2] The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path
The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.
Suppose that we would like to estimate For that we have generated two samples An unbiased estimate of
is given by And so variance is reduced if
is negative.
If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be
is obtained from U(0, 1).
The second sample is built from
Furthermore, covariance is negative, allowing for initial variance reduction.
We would like to estimate The exact result is
This integral can be seen as the expected value of
, where 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.