Last month, Roberto Casarin, Radu Craiu, Lorenzo Frattarolo and myself posted an arXiv paper on a unified approach to antithetic sampling. To which I mostly and modestly contributed while visiting Roberto in Venezia two years ago (although it seems much farther than that!). I have always found antithetic sampling fascinating, albeit mostly unachievable in realistic situations, except (and approximately) by quasi-random tools. The original approach dates back to Hammersley and Morton, circa 1956, when they optimally couple X=F⁻(U) and Y=F⁻(1-U), with U Uniform, although there is no clear-cut extension beyond pairs or above dimension one. While the search for optimal and feasible antithetic plans dried out in the mid-1980’s, despite near successes by Rubinstein and others, the focus switched to Latin hypercube sampling.
The construction of a general antithetic sampling scheme is based on sampling uniformly an edge within an undirected graph in the d-dimensional hypercube, under some (three) assumptions on the edges to achieve uniformity for the marginals. This construction achieves the smallest Kullback-Leibler divergence between the resulting joint and the product of uniforms. And it can be furthermore constrained to be d-countermonotonic, ie such that a non-linear sum of the components is constant. We also show that the proposal leads to closed-form Kendall’s τ and Spearman’s ρ. Which can be used to assess different d-countermonotonic schemes, incl. earlier ones found in the literature. The antithetic sampling proposal can be applied in Monte Carlo, Markov chain Monte Carlo, and sequential Monte Carlo settings. In a stochastic volatility example of the later (SMC) we achieve performances similar to the quasi-Monte Carlo approach of Mathieu Gerber and Nicolas Chopin.
Xi'an's Og 