Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-14T10:06:20.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Monte Carlo integration with a growing number of control variates

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

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, D. W. K. (1991). Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrika 59, 307–345.CrossRefGoogle Scholar
Bardenet, R. and Hardy, A. (2016). Monte Carlo with determinantal point processes. arXiv:1605.00361.Google Scholar
Brass, H. and Petras, K. (2011). Quadrature Theory (Math. Surv. Monogr. 178). American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Bugeaud, Y. (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press.CrossRefGoogle Scholar
Delyon, B. and Portier, F. (2016). Integral approximation by kernel smoothing. Bernoulli 22, 2177–2208.CrossRefGoogle Scholar
Dick, J. and Pillichshammer, F. (2010). Digital Nets and Sequences. Cambridge University Press.CrossRefGoogle Scholar
Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.CrossRefGoogle Scholar
Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York.CrossRefGoogle Scholar
Glasserman, P. and Yu, B. (2005). Large sample properties of weighted Monte Carlo estimators. Operat. Res. 53, 298–312.CrossRefGoogle Scholar
Glynn, P. W. and Szechtman, R. (2002). Some new perspectives on the method of control variates. In Monte Carlo and Quasi-Monte Carlo Methods, 2000 (Hong Kong). Springer, Berlin, pp. 27–49.CrossRefGoogle Scholar
Gobet, E. and Labart, C. (2010). Solving BSDE with adaptive control variate. SIAM J. Numer. Anal. 48, 257–277.CrossRefGoogle Scholar
Hesterberg, T. and Nelson, B. (1998). Control variates for probability and quantile estimation. Manag. Sci. 44, 1295–1312.CrossRefGoogle Scholar
Huber, P. J. (1981). Robust Statistics. John Wiley, New York.CrossRefGoogle Scholar
Jie, T. and Abbeel, P. (2010). On a connection between importance sampling and the likelihood ratio policy gradient. In Advances in Neural Information Processing Systems 23, eds Lafferty, J. D., Williams, C. K. I., Shawe-Taylor, J., Zemel, R. S. and Culotta, A., pp. 1000–1008.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Leluc, R., Portier, F. and Segers, J. (June 2019). Control variates selection for Monte Carlo integration. arXiv:1906.10920.Google Scholar
Lorentz, G. G. (1986). Approximation of Functions, 2nd edn. Chelsea Publishing Co., New York.Google Scholar
McCulloch, C. E. and Searle, S. R. (2001). Generalized, Linear, and Mixed Models. Wiley-Interscience, New York.Google Scholar
McFadden, D. (2001). Economic choices. Amer. Rconom. Rev. 91, 351–378.Google Scholar
Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79, 147–168.CrossRefGoogle Scholar
Novak, E. (2016). Some results on the complexity of numerical integration. In Monte Carlo and Quasi-Monte Carlo Methods. Springer, Berlin, pp. 161–183.CrossRefGoogle Scholar
Oates, C. J., Cockayne, J., Briol, F.-X. and Girolami, M. (2018). Convergence rates for a class of estimators based on Stein’s method. To appear in Bernoulli.Google Scholar
Oates, C. J. and Girolami, M. (2016). Control functionals for quasi-Monte Carlo integration. arXiv:1501.03379v7.Google Scholar
Oates, C. J., Girolami, M. and Chopin, N. (2017). Control functionals for Monte Carlo integration. J. R. Statist. Soc. B 79, 695–718.CrossRefGoogle Scholar
Owen, A. B. (2013). Monte Carlo theory, methods and examples. Available at http://statweb.stanford.edu/~owen/mc/.Google Scholar
Owen, A. and Zhou, Y. (2000). Safe and effective importance sampling. J. Amer. Statist. Assoc. 95, 135–143.CrossRefGoogle Scholar
Portier, F. and Delyon, B. (2018). Asymptotic optimality of adaptive importance sampling. In Advances in Neural Information Processing Systems 18, pp. 3134–3144.Google Scholar
Portier, F. and Segers, J. (March 2018). Monte Carlo integration with a growing number of control variates. arXiv:1801.01797v3.Google Scholar
Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer Texts in Statistics. Springer, New York.CrossRefGoogle Scholar
South, L. F., Oates, C. J., Mira, A. and Drovandi, C. (2018). Regularised zero-variance control variates. arXiv:1811.05073.Google Scholar
Velleman, P. F. and Welsch, R. E. (1981). Efficient computing of regression diagnostics. Amer. Statistician 35, 234–242.Google Scholar
Wang, H. and Xiang, S. (2012). On the convergence rates of Legendre approximation. Math. Comput. 81, 861–877.CrossRefGoogle Scholar
Zhang, P. (1996). Nonparametric importance sampling. J. Amer. Statist. Assoc. 91, 1245–1253.CrossRefGoogle Scholar