{"articles":{"antithetic-variates-on-monotonic@computational-statistical-methods":{"content":"<p>Let \\(h\\) be monotonic. <br> One way to implement antithetic sampling is to exploit the fact that \\(h\\) is monotonic. We can easily generate negatively correlated uniform variables, and monotonicity preserves this negative correlation when applied to the variables. <br> Sample from \\(U\\) to get \\({X}_{i}\\) and \\({{Y}_{i} = 1 - {X}_{i}}\\), two opposite values on \\({{\\left[0..1\\right]}}\\). \\({X}_{i}\\) and \\({Y}_{i}\\) are negatively correlated, and since \\(h\\) is monotonic, \\({h({X}_{i})}\\) and \\({h({Y}_{i})}\\) are as well. <br> The Monte Carlo estimator is \\({{\\hat\\mu}_{\\text{AS}} = {\\sum}_{i} \\frac{h(X_i) + h(Y_i)}{2}}\\).</p>","names":[[["Antithetic variates on monotonic function",""]]]}},"style":"Method"}