Antithetic variates explained

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

\{\varepsilon _{1,...,\varepsilon} _{M\}} to also take \{ - \varepsilon _{1,...,-\varepsilon} _{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

\theta = E ( h(X) ) = E ( Y )

For that we have generated two samples

Y _{1andY} _{2}

An unbiased estimate of

{\theta} is given by\hat \theta = (Y _{1} + Y _{2})/(2) .

And

Var (\hat \theta) = (Var (Y _{1)} + Var (Y _{2)} +2 Cov (Y _{1,Y} _{2)})/(4)

so variance is reduced if

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

u _{1,} \ldots, u _{n} , where, for any given i, u _{i} is obtained from U(0, 1). The second sample is built from u' _{1,} \ldots, u' _{n} , where, for any given i: u' _{i} = 1- u _{i} . If the set u _{i} 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 = \int ^{1} _{0} (1)/(1+ x) d x.

The exact result is

I =ln 2 ≈ 0 . 69314718 . This integral can be seen as the expected value of f(U) , wheref(x) = (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

Notes and References

1. Botev. Z.. Ridder. A.. Variance Reduction. Wiley StatsRef: Statistics Reference Online. 2017. 1–6. 10.1002/9781118445112.stat07975. 9781118445112.
2. Book: Kroese. D. P.. Dirk Kroese . Taimre. T.. Botev. Z. I.. Handbook of Monte Carlo methods. 2011 . John Wiley & Sons. (Chapter 9.3)