# 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.[1][2]

## Underlying principle

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path $\displaystyle{ \{\varepsilon_1,\dots,\varepsilon_M\} }$ to also take $\displaystyle{ \{-\varepsilon_1,\dots,-\varepsilon_M\} }$. 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

$\displaystyle{ \theta = \mathrm{E}( h(X) ) = \mathrm{E}( Y ) \, }$

For that we have generated two samples

$\displaystyle{ Y_1\text{ and }Y_2 \, }$

An unbiased estimate of $\displaystyle{ {\theta} }$ is given by

$\displaystyle{ \hat \theta = \frac{Y_1 + Y_2}{2}. }$

And

$\displaystyle{ \text{Var}(\hat \theta) = \frac{\text{Var}(Y_1) + \text{Var}(Y_2) + 2\text{Cov}(Y_1,Y_2)}{4} }$

so variance is reduced if $\displaystyle{ \text{Cov}(Y_1,Y_2) }$ is negative.

## Example 1

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be $\displaystyle{ u_1, \ldots, u_n }$, where, for any given i, $\displaystyle{ u_i }$ is obtained from U(0, 1). The second sample is built from $\displaystyle{ u'_1, \ldots, u'_n }$, where, for any given i: $\displaystyle{ u'_i = 1-u_i }$. If the set $\displaystyle{ u_i }$ is uniform along [0, 1], so are $\displaystyle{ u'_i }$. 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 − 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.

## References

1. Botev, Z.; Ridder, A. (2017). "Variance Reduction". Wiley StatsRef: Statistics Reference Online: 1–6. doi:10.1002/9781118445112.stat07975. ISBN 9781118445112.
2. Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons. (Chapter 9.3)