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Monte Carlo integration with a growing number of control variates

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems Linear inference, regression

Published online by Cambridge University Press:  11 December 2019

FranÇois Portier Open the ORCID record for FranÇois Portier [Opens in a new window]  and
Johan Segers Open the ORCID record for Johan Segers [Opens in a new window]
FranÇois Portier*
Affiliation:
Télécom Paris
Johan Segers*
Affiliation:
UCLouvain
*
*Postal address: LTCI, Télécom Paris, Institut Polytechnique de Paris, Rue Barrault, 75013 Paris, France.
***Postal address: Institut de Statistique, Biostatistique et Sciences Actuarialles, LIDAM, UCLouvain, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium.
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Abstract

It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem is established for the integration error if the number of control variates tends to infinity. The integration error is scaled by the standard deviation of the error term in the regression model. If the linear span of the control variates is dense in a function space that contains the integrand, the integration error tends to zero at a rate which is faster than the square root of the number of Monte Carlo replicates. Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.

Keywords

Central limit theoremcontrol variatesmultiple linear regressionordinary least squarespost-stratificationLegendre polynomial

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62J05: Linear regression 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1168 - 1186
Copyright
© Applied Probability Trust 2019 

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