Antithetic variates

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]

## Underlying principle

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The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path -- that is given a path  { ε 1 , … , ε M } {\displaystyle \\{\varepsilon _{1},\dots ,\varepsilon _{M}\\}} to also take  { − ε 1 , … , − ε M } {\displaystyle \\{-\varepsilon _{1},\dots ,-\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

θ = E ( h ( X ) ) = E ( Y ) {\displaystyle \theta =\mathrm {E} (h(X))=\mathrm {E} (Y)\,}

For that we have generated two samples

Y 1 and Y 2 {\displaystyle Y_{1}{\text{ and }}Y_{2}\,}

An unbiased estimate of  θ {\displaystyle {\theta }} is given by

θ ^ = Y 1 + Y 2 2 . {\displaystyle {\hat {\theta }}={\frac {Y_{1}+Y_{2}}{2}}.}

And

Var ( θ ^ ) = Var ( Y 1 ) + Var ( Y 2 ) + 2 Cov ( Y 1 , Y 2 ) 4 {\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  Cov ( Y 1 , Y 2 ) {\displaystyle {\text{Cov}}(Y_{1},Y_{2})} is negative.

## Example 1

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If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be  u 1 , … , u n {\displaystyle u_{1},\ldots ,u_{n}} , where, for any given i,  u i {\displaystyle u_{i}} is obtained from U(0, 1). The second sample is built from  u 1 ′ , … , u n ′ {\displaystyle u'_{1},\ldots ,u'_{n}} , where, for any given i:  u i ′ = 1 − u i {\displaystyle u'_{i}=1-u_{i}} . If the set  u i {\displaystyle u_{i}} is uniform along [0, 1], so are  u i ′ {\displaystyle u'_{i}} . Furthermore, covariance is negative, allowing for initial variance reduction.

## Example 2: integral calculation

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We would like to estimate

I = ∫ 0 1 1 1 + x d x . {\displaystyle I=\int _{0}^{1}{\frac {1}{1+x}}\,\mathrm {d} x.}

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

f ( x ) = 1 1 + x {\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 − 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

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* Control variates

## References

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1. ^ Botev, Z.; Ridder, A. (2017). "Variance Reduction". Wiley StatsRef: Statistics Reference Online. pp. 1–6\. doi:10.1002/9781118445112.stat07975. hdl:1959.4/unsworks_50616. ISBN 9781118445112.
2. ^ Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons.(Chapter 9.3)