Antithetic variates

Short description: Monte Carlo method

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 {ε1,…,εM} to also take {−ε1,…,−εM}. 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

θ=E(h(X))=E(Y)

For that we have generated two samples

Y1 and Y2

An unbiased estimate of θ is given by

θ^=Y1+Y22.

And

Var(θ^)=Var(Y1)+Var(Y2)+2Cov(Y1,Y2)4

so variance is reduced if Cov(Y1,Y2) is negative.

## Example 1

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be u1,…,un, where, for any given i, ui is obtained from U(0, 1). The second sample is built from u'1,…,u'n, where, for any given i: u'i=1−ui. If the set ui 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=∫0111+xdx.

The exact result is I=ln⁡2≈0.69314718. This integral can be seen as the expected value of f(U), where

f(x)=11+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 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. pp. 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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