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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]

Contents

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 accuracy.

Suppose that we would like to estimate

 

For that we have generated two samples

 

An unbiased estimate of   is given by

 

And

 

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 Variance
Classical Estimate 0.69365 0.002005
Antithetic Variates 0.69399 0.00063

The use of the antithetic variates method to estimate the result shows an important variance reduction.

ReferencesEdit

  1. ^ Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons. (Chapter 9.3)