Abstract
We identify recurrent ingredients in the antithetic sampling literature leading to a unified sampling framework. We introduce a new class of antithetic schemes that includes the most used antithetic proposals. This perspective enables the derivation of new properties of the sampling schemes: (i) optimality in the Kullback–Leibler sense; (ii) closed-form multivariate Kendall’s τ and Spearman’s ρ; (iii) ranking in concordance order and (iv) a central limit theorem that characterizes stochastic behaviour of Monte Carlo estimators when the sample size tends to infinity. The proposed simulation framework inherits the simplicity of the standard antithetic sampling method, requiring the definition of a set of reference points in the sampling space and the generation of uniform numbers on the segments joining the points. We provide applications to Monte Carlo integration and Markov Chain Monte Carlo Bayesian estimation.
Funding Statement
Radu V. Craiu’s research was supported by NSERC of Canada grant RGPIN-249547. Roberto Casarin acknowledges the support from: the Venice Centre for Risk Analytics (VERA) at the Ca’ Foscari University of Venice; the European Union – Next Generation EU – under both the project ‘GRINS – Growing Resilient, INclusive and Sustainable’, reference (PE0000018), the National Recovery and Resilience Plan (NRRP) PE9, as well as the MUR - Prin 2022 project ‘Discrete random structures for Bayesian learning and prediction’, Grant no. 2022CLTYP4. 4. Christian P. Robert’s research has been supported by the French Government under both a senior chair (2016-2021) from l’Institut Universitaire de France and under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR19-P3IA-0001 (PRAIRIE 3IA Institute), as well as the ERC Synergy grant 101071601 (OCEAN).
Acknowledgments
The views expressed by Lorenzo Frattarolo are the author’s alone and do not necessarily correspond to those of the European Commission. We thank for their useful comments the participants at: the 1st Italian-French Statistics Workshop, October 2017, Italy; the 32nd European Statisticians Meeting, Italy, July 2019; the Workshop “Monte Carlo methods, and approximate dynamic programming”, France, October 2019; the XXII Workshop on Quantitative Finance, Italy, April 2022; the 49th Annual Meeting of the Statistical Society of Canada, May 2022; the XXXII Conference of the Association of European Operational Research Societies, Finland, July 2022; the 24th International Conference on Computational Statistics, Bologna, Italy, August 2022; and the 12th European Bayesian Econometrics Seminar, Salzburg, Austria, September 2022.
Citation
Roberto Casarin. Radu V. Craiu. Lorenzo Frattarolo. Christian P. Robert. "Living on the Edge: An Unified Approach to Antithetic Sampling." Statist. Sci. 39 (1) 115 - 136, February 2024. https://doi.org/10.1214/23-STS888
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