Antithetic variates deliberately couple unbiased estimators of the same expectation with negative covariance. Their average can have smaller variance than an average using independent draws. For example, evaluating a monotone integrand at and , where has a uniform distribution, gives opposite fluctuations while preserving both marginal distributions.
For standard Cauchy distribution probability density function and , this estimates its upper-tail probability without sampling a heavy tail. The function decreases on , while increases, so their covariance is nonpositive. The paired estimator is unbiased and has no larger variance than an estimator using the same number of independent probability density function evaluations.
For two unbiased estimators of the same parameter with variances and covariance , the fixed-weight estimator is unbiased and has variance . Completing the square gives and minimum variance when the denominator is positive. These are optimal among fixed linear unbiased combinations. Substituting weights estimated from the same observations need not preserve unbiasedness.
Split independent observational blocks into two folds. Use one fold to estimate the combination weight for the two sample means in the other fold, and conversely. Each held-out combination is an unbiased estimator conditionally on its training fold because its two held-out sample means have the same expectation. A deterministic average of the two combinations remains unbiased. Under consistency of the moment estimates and balanced fold sizes, the weights approach those of the minimum-variance combination of unbiased sample means. Paired observations must stay together when splitting or bootstrapping so their covariance is preserved.
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Antithetic variates is a variance reduction technique used in the context of Monte Carlo simulation. The main purpose of this technique is to improve the efficiency of the simulation by reducing the variance of the estimator. The idea behind antithetic variates is to generate pairs of dependent random variables that are negatively correlated. This negation helps to balance out the fluctuations that might occur in the estimated outcomes.