# Antithetic variates

In statistics, the **antithetic variates** method is a variance reduction technique used in Monte Carlo methods. Considering that the error reduction in the simulated signal (using Monte Carlo methods) has a 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 principleEdit

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path to also take . 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.

## Example 1Edit

If the law of the variable *X* follows a uniform distribution along [0, 1], the first sample will be , where, for any given *i*, is obtained from *U*(0, 1). The second sample is built from , where, for any given *i*: . If the set is uniform along [0, 1], so are . Furthermore, covariance is negative, allowing for initial variance reduction.

## Example 2: integral calculationEdit

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: 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.

## ReferencesEdit

**^**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)