# Antithetic variates

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.

## Underlying principle

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path $\{\varepsilon _{1},\dots ,\varepsilon _{M}\}$ to also take $\{-\varepsilon _{1},\dots ,-\varepsilon _{M}\}$ . 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

$\theta =\mathrm {E} (h(X))=\mathrm {E} (Y)\,$ For that we have generated two samples

$Y_{1}{\text{ and }}Y_{2}\,$ An unbiased estimate of ${\theta }$ is given by

${\hat {\theta }}={\frac {Y_{1}+Y_{2}}{2}}.$ And

${\text{Var}}({\hat {\theta }})={\frac {{\text{Var}}(Y_{1})+{\text{Var}}(Y_{2})+2{\text{Cov}}(Y_{1},Y_{2})}{4}}$ so variance is reduced if ${\text{Cov}}(Y_{1},Y_{2})$ is negative.

## Example 1

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

## Example 2: integral calculation

We would like to estimate

$I=\int _{0}^{1}{\frac {1}{1+x}}\,\mathrm {d} x.$ The exact result is $I=\ln 2\approx 0.69314718$ . This integral can be seen as the expected value of $f(U)$ , where

$f(x)={\frac {1}{1+x}}$ 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 − ui):

 Estimate Standard deviation Classical Estimate 0.69365 0.00255 Antithetic Variates 0.69399 0.00063

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