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]

## Contents

* 1 Underlying principle
* 2 Example 1
* 3 Example 2: integral calculation
* 4 See also
* 5 References

## 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: 2n, where n = 1500) to the antithetic variates estimate (sample size: n, completed with the transformed sample 1 − ui):

| Estimate | standard error

|
|

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

* Control variates

## 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)

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