Antithetic variates

Remarks

Clause 1 holds because $1 - U$ is uniform on $(0, 1)$ too. For clause 2, if $f$ is non-decreasing then $u \mapsto f(1 - u)$ is non-increasing, and for a non-decreasing $a$ and a non-increasing $b$ one has $\mathbb E[a(U) b(U)] \le \mathbb E a(U)\, \mathbb E b(U)$, the integral form of Chebyshev's sum inequality. This in turn follows from $\mathbb E\bigl[(a(U) - a(V))(b(U) - b(V))\bigr] \le 0$ for an independent copy $V$ of $U$, since the two factors have opposite signs. Clause 3: $\operatorname{Var} Z = \frac14 \bigl( 2 \operatorname{Var} f(U) + 2 \operatorname{Cov}(f(U), f(1-U)) \bigr) \le \frac12 \operatorname{Var} f(U)$, using $\operatorname{Var} f(1 - U) = \operatorname{Var} f(U)$. Example: for $f(u) = e^u$, $\operatorname{Var} f(U) \approx 0.2420$ while $\operatorname{Var} Z \approx 0.0039$, a gain of a factor of about $30$ over $2$ independent samples. Monotonicity is needed: for $f(u) = (u - \frac12)^2$ one has $f(1 - u) = f(u)$, so $Z = f(U)$ and $\operatorname{Var} Z = \operatorname{Var} f(U)$, twice the variance of the average of two independent samples.

Going deeper

For a student

Antithetic variates is one of the simplest variance reduction techniques, with no extra modelling and almost no extra cost. The same idea applies to normal inputs with the pair $Z$ and $-Z$, and to simulations driven by many uniforms, replacing each $U_j$ by $1 - U_j$. In the multivariate case the guarantee holds when the output is a monotone function of each input separately, but for general outputs the method can increase the variance. Combined with inverse transform sampling, which is monotone, it yields negatively correlated pairs from any distribution.