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




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

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.


  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)