antithetic variates

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path $\backslash \{\backslash varepsilon\_1,\backslash dots,\backslash varepsilon\_M\backslash \}$ to also take $\backslash \{-\backslash varepsilon\_1,\backslash dots,-\backslash varepsilon\_M\backslash \}$. 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

:$\backslash theta\; =\; \backslash mathrm\{E\}(\; h(X)\; )\; =\; \backslash mathrm\{E\}(\; Y\; )\; \backslash ,$

For that we have generated two samples

:$Y\_1\backslash text\{\; and\; \}Y\_2\; \backslash ,$

An unbiased estimate of $\{\backslash theta\}$ is given by

:$\backslash hat\; \backslash theta\; =\; \backslash frac\{Y\_1\; +\; Y\_2\}\{2\}.$

And

:$\backslash text\{Var\}(\backslash hat\; \backslash theta)\; =\; \backslash frac\{\backslash text\{Var\}(Y\_1)\; +\; \backslash text\{Var\}(Y\_2)\; +\; 2\backslash text\{Cov\}(Y\_1,Y\_2)\}\{4\}$

so variance is reduced if $\backslash text\{Cov\}(Y\_1,Y\_2)$ is negative.

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be $u\_1,\; \backslash ldots,\; u\_n$, where, for any given i, $u\_i$ is obtained from U(0, 1). The second sample is built from $u\text{'}\_1,\; \backslash ldots,\; u\text{'}\_n$, where, for any given i: $u\text{'}\_i\; =\; 1-u\_i$. If the set $u\_i$ is uniform along [0, 1], so are $u\text{'}\_i$. Furthermore, covariance is negative, allowing for initial variance reduction.

We would like to estimate

:$I\; =\; \backslash int\_0^1\; \backslash frac\{1\}\{1+x\}\; \backslash ,\; \backslash mathrm\{d\}x.$

The exact result is $I=\backslash ln\; 2\; \backslash approx\; 0.69314718$. This integral can be seen as the expected value of $f(U)$, where

:$f(x)\; =\; \backslash frac\{1\}\{1+x\}$

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):

:

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

• Control variates

{{Reflist}}

Category:Variance reduction

Category:Computational statistics

Category:Monte Carlo methods