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Combination of General Antithetic Transformations and Control Variables

GERAD and Département des méthodes quantitatives de gestion, HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, Montréal (Québec) Canada H3T 2A7
GERAD and Département d'informatique et de recherche opérationnelle, Université de Montréal, C.P. 6128, succ. Centre-Ville, Montréal, Quebec, Canada
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4
hatem.ben-ameur{at}hec.ca
www.iro.umontreal.ca/~lecuyer
lemieux{at}math.ucalgary.ca

Several methods for reducing the variance in the context of Monte Carlo simulation are based on correlation induction. This includes antithetic variates, Latin hypercube sampling, and randomized version of quasi-Monte Carlo methods such as lattice rules and digital nets, where the resulting estimators are usually weighted averages of several dependent random variables that can be seen as function evaluations at a finite set of random points in the unit hypercube. In this paper, we consider a setting where these methods can be combined with the use of control variates and we provide conditions under which we can formally prove that the variance is minimized by choosing equal weights and equal control variate coefficients across the different points of evaluation, regardless of the function (integrand) that is evaluated.

Key Words: variance reduction; control variates; antithetic variates; quasi-Monte Carlo; lattice rules; digital nets; Latin hypercube sampling
History: Received: March 25, 2003; revision received: October 16, 2003;

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