Antithetic Variates

Concept Overview

The method of antithetic variates is a variance reduction technique used in Monte Carlo simulation. When we want to estimate the expected value of a function f(U) where U is a uniform random variable on [0,1], we can generate pairs of inputs rather than independent samples. For a monotone function with finite variance, the pair (U, 1-U) gives nonpositive covariance between f(U) and f(1-U). The pair average (f(U) + f(1-U))/2 therefore has no greater variance than the average of two independent evaluations; constant functions can give equality.

Worked Example

Let's consider estimating the expected value of the function f(u) = u^2 using the method of antithetic variates.

Set Probe eighths to 3 to inspect the illustrative input u = 3/8 (which is 0.375).

First evaluation: We calculate f(u) = (3/8)^2 = 9/64 = 0.140625.
Antithetic counterpart: We construct the mirrored input 1 - u = 5/8 (or 0.625).
Second evaluation: We calculate f(1-u) = (5/8)^2 = 25/64 = 0.390625.
Pair average: The antithetic estimator for this pair is (9/64 + 25/64) / 2 = 34/128 = 17/64 = 0.265625.

The true mean of u^2 on [0,1] is 1/3. The variance of the antithetic estimator is calculated across the entire uniform distribution. The variance of two independent evaluations is 2/45 ≈ 0.0444, while the variance of the antithetic pair is 1/180 ≈ 0.0055. Because u^2 is strictly monotonic on [0,1], the antithetic approach successfully reduces the estimator variance.

Experiment

You can use the Antithetic Variates interactive module to visualize how the mirrored pair is constructed for different functions and observe the effect on the estimator's variance.

Set the Function preset to Square.
Move the Probe eighths slider to 3 to represent the uniform sample u = 3/8.
Observe the canvas where the evaluations f(U) and f(Mirror) are plotted, and the pair average falls exactly in the middle.
Examine the Estimator variances panel. Compare the variance of the independent estimator (two independent uniform samples) with the variance of the antithetic pair. Notice that for the Square function, the antithetic variance is significantly lower.
Change the Function preset to Folded absolute. You will see that the antithetic estimator now has a higher variance (1/48) compared to the independent estimator (1/96), because this function is not monotonic and its structure defeats the negative correlation strategy.

Limitations and Boundaries

The effectiveness of antithetic variates strongly depends on the function being integrated:

Monotonicity is key: For a monotone function with finite variance, the technique guarantees no higher variance than independent pairing; strict improvement requires negative covariance. For example, a nonconstant linear function yields a perfect negative correlation, reducing the variance to zero.
Non-monotonic functions can perform worse: If the function is symmetric around the midpoint (like abs(u - 0.5)), f(u) and f(1-u) will be perfectly positively correlated. The pair average will then have twice the variance of an estimator that takes two independent samples.
Displayed variances: The variances shown in the module are exact analytical results integrated over the entire uniform law, not Monte Carlo estimates derived from the single illustrated pair.

References

The following reference describes the principles of variance reduction techniques in Monte Carlo simulation, providing the mathematical background for this module's design.

STAT 7400: Computer Intensive Statistics - Simulation
Authored by Luke Tierney at the University of Iowa, these notes discuss variance reduction techniques including antithetic covariance and control variates.

Antithetic Variates