# 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 [math]\displaystyle{ \{\varepsilon_1,\dots,\varepsilon_M\} }[/math] to also take [math]\displaystyle{ \{-\varepsilon_1,\dots,-\varepsilon_M\} }[/math]. 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

- [math]\displaystyle{ \theta = \mathrm{E}( h(X) ) = \mathrm{E}( Y ) \, }[/math]

For that we have generated two samples

- [math]\displaystyle{ Y_1\text{ and }Y_2 \, }[/math]

An unbiased estimate of [math]\displaystyle{ {\theta} }[/math] is given by

- [math]\displaystyle{ \hat \theta = \frac{Y_1 + Y_2}{2}. }[/math]

And

- [math]\displaystyle{ \text{Var}(\hat \theta) = \frac{\text{Var}(Y_1) + \text{Var}(Y_2) + 2\text{Cov}(Y_1,Y_2)}{4} }[/math]

so variance is reduced if [math]\displaystyle{ \text{Cov}(Y_1,Y_2) }[/math] is negative.

## Example 1

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

## Example 2: integral calculation

We would like to estimate

- [math]\displaystyle{ I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x. }[/math]

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

- [math]\displaystyle{ f(x) = \frac{1}{1+x} }[/math]

and *U* follows a uniform distribution [0, 1].

The following table compares the classical Monte Carlo estimate (sample size: 2*n*, where *n* = 1500) to the antithetic variates estimate (sample size: *n*, completed with the transformed sample 1 − *u*_{i}):

**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.

## See also

## References

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

Original source: https://en.wikipedia.org/wiki/Antithetic variates.
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