Antithetic variates

Example 1

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be ${\displaystyle u_1,\dots ,u_n}$, where, for any given i, ${\displaystyle u_i}$ is obtained from U(0, 1). The second sample is built from ${\displaystyle u_1^{\prime },\dots ,u_n^{\prime }}$, where, for any given i: ${\displaystyle u_i^{\prime }=1-u_i}$. If the set ${\displaystyle u_i}$ is uniform along [0, 1], so are ${\displaystyle u_i^{\prime }}$. Furthermore, covariance is negative, allowing for initial variance reduction.

Example 2: integral calculation

We would like to estimate

${\displaystyle I={\int }_0^1\frac{1}{1+x}\mathrm{d}x.}$

The exact result is ${\displaystyle I=\ln 2\approx 0.69314718}$. This integral can be seen as the expected value of ${\displaystyle f(U)}$, where

${\displaystyle f(x)=\frac{1}{1+x}}$

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 − u_i):

The use of the antithetic variates method to estimate the result shows an important variance reduction.