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 ${\displaystyle \{{\epsilon }_1,\dots ,{\epsilon }_M\}}$ to also take ${\displaystyle \{-{\epsilon }_1,\dots ,-{\epsilon }_M\}}$. 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 ${\displaystyle u_1,\dots ,u_n}$, where, for any given i, ${\displaystyle u_i}$ is obtained from U(0, 1). The second sample is built from ${\displaystyle u_1^{\prime },\dots ,u_n^{\prime }}$, where, for any given i: ${\displaystyle u_i^{\prime }=1-u_i}$. If the set ${\displaystyle u_i}$ is uniform along [0, 1], so are ${\displaystyle u_i^{\prime }}$. 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 − u_i):

	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

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