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Antithetic variates preserve each draw's marginal law but couple two estimates negatively. For an average of identically distributed $H,H^{\prime }$,Negative covariance improves on two independent evaluations at the same computational budget.

Here generate independent $U_1,\dots ,U_r\sim U(0,1)$ and pair $x_j=k{U}_j$ with $x_j^{\prime }=k(1-U_j)$. Both are uniform on $[0,k]$. The antithetic Cauchy-tail integration on a finite interval estimator isIt is unbiased. Put $h(u)=f(ku)$. This decreases, whereas $h(1-u)$ increases. For an independent copy $U^{\prime }$ of $U$, the covariance identity givesEach product is nonpositive, and for $k>0$ it is strictly negative off the diagonal. Thereforewhich is the variance using $2r$ independent evaluations in part (v). Thus the comparison accounts for the doubled number of probability density function evaluations in each pair.