Abstract
We propose randomized quasi-Monte Carlo (RQMC) methods to estimate expectations \(\mu = {\mathbb {E}}(g(\boldsymbol{Y}, W))\) where \(\boldsymbol{Y}\) is independent of W and can be sampled by inversion, whereas W cannot. Various practical problems are of this form, such as estimating expected shortfall for mixture models where W is stable or generalized inverse Gaussian and \(\boldsymbol{Y}\) is multivariate normal. We consider two settings: In the first, we assume that there is a non-uniform random variate generation method to sample W in the form of a non-modifiable “black-box”. The methods we propose for this setting are based on approximations of the quantile function of W. In the second setting, we assume that there is an acceptance-rejection (AR) algorithm to sample from W and explore different ways to feed it with quasi-random numbers. This has been studied previously, typically by rejecting points of constant dimension from a low-discrepancy sequence and moving along the sequence. We also investigate the use of a point set of constant (target) size where the dimension of each point is increased until acceptance. In addition, we show how to combine the methods from the two settings in such a way that the non-monotonicity inherent to AR is removed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Chambers, J., Mallows, C., Stuck, B.: A method for simulating stable random variables. J. Amer. Stat. Assoc. 71(354), 340–344 (1976). https://doi.org/10.1080/01621459.1976.10480344
Cheng, R.: The generation of Gamma variables with non-integral shape parameter. J. Roy. Stat. Soc.: Ser. C (Appl. Stat.) 26(1), 71–75 (1977)
Wuertz, D., Maechler, M., Rmetrics core team members: Stabledist: Stable Distribution Functions (2016). https://CRAN.R-project.org/package=stabledist. R package version 0.7-1
Demarta, S., McNeil, A.: The t copula and related copulas. Int. Stat. Rev. 73(1), 111–129 (2005). https://doi.org/10.1111/j.1751-5823.2005.tb00254.x
Derflinger, G., Hörmann, W., Leydold, J.: Random variate generation by numerical inversion when only the density is known. ACM Trans. Model. Comput. Simul. (TOMACS) 20(4), 1–25 (2010)
Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986). https://doi.org/10.1007/978-1-4613-8643-8
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling extremal events. Br. Actuar. J. 5(2), 465–465 (1999)
Flury, B.: Acceptance-rejection sampling made easy. SIAM Rev. 32(3), 474–476 (1990)
Glasserman, P.: Monte Carlo Methods in Financial Engineering, vol. 53. Springer Science & Business Media, Berlin (2013)
Hartinger, J., Kainhofer, R.: Non-uniform low-discrepancy sequence generation and integration of singular integrands. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 163–179. Springer, Berlin (2006)
Hintz, E., Hofert, M., Lemieux, C.: Grouped normal variance mixtures. Risks 8(4), 103 (2020). https://doi.org/10.3390/risks8040103
Hintz, E., Hofert, M., Lemieux, C.: Normal variance mixtures: distribution, density and parameter estimation. Comput. Stat. Data Anal. 157C, 107175 (2021). https://doi.org/10.1016/j.csda.2021.107175
Hofert, M., Lemieux, C.: qrng: (Randomized) Quasi-Random Number Generators (2019). https://CRAN.R-project.org/package=qrng. R package version 0.0-7
Hörmann, W., Leydold, J.: Generating generalized inverse Gaussian random variates. Stat. Comput. 24(4), 547–557 (2014). https://doi.org/10.1007/s11222-013-9387-3
Kundu, D., Gupta, R.: A convenient way of generating Gamma random variables using generalized exponential distribution. Comput. Stat. Data Anal. 51(6), 2796–2802 (2007). https://doi.org/10.1016/j.csda.2006.09.037
L’Ecuyer, P.: Quasi-Monte Carlo methods in finance. In: Proceedings of the 2004 Winter Simulation Conference, vol. 2, pp. 1645–1655. IEEE (2004)
L’Ecuyer, P., Lécot, C., Tuffin, B.: A randomized quasi-Monte Carlo simulation method for Markov chains. Oper. Res. 56(4), 958–975 (2008)
L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Manage. Sci. 46(9), 1214–1235 (2000)
L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszki, F. (eds.) Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp. 419–474. Kluwer Academic Publishers, Boston (2002)
L’Ecuyer, P., Munger, D., Lécot, C., Tuffin, B.: Sorting methods and convergence rates for array-RQMC: some empirical comparisons. Math. Comput. Simul. 143, 191–201 (2018)
Leydold, J., Hörmann, W.: Generating generalized inverse Gaussian random variates by fast inversion. Comput. Stat. & Data Anal. 55(1), 213–217 (2011)
Leydold, J., Hörmann, W.: Runuran: R Interface to the ’UNU.RAN’ Random Variate Generators (2020). https://CRAN.R-project.org/package=Runuran. R package version 0.30
McNeil, A., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts. Techniques and Tools. Princeton University Press (2015). https://doi.org/10.1007/s10687-017-0286-4
Moskowitz, B., Caflisch, R.: Smoothness and dimension reduction in quasi-Monte Carlo methods. Math. Comput. Model. 23(8–9), 37–54 (1996). https://doi.org/10.1016/0895-7177(96)00038-6
Nakayama, M., Kaplan, Z.T., L’Ecuyer, P., Tuffin, B.: Quantile estimation via a combination of conditional Monte Carlo and randomized quasi-Monte Carlo. In: Proceedings of the 2020 Winter Simulation Conference (2020)
Nguyen, N., Ökten, G.: The acceptance-rejection method for low-discrepancy sequences. Monte Carlo Methods Appl. 22(2), 133–148 (2016). https://doi.org/10.1515/mcma-2016-0104
Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23(3), 470–472 (1952). https://doi.org/10.1214/aoms/1177729394
Wang, X.: Improving the rejection sampling method in quasi-Monte Carlo methods. J. Comput. Appl. Math. 114(2), 231–246 (2000). https://doi.org/10.1016/S0377-0427(99)00194-6
Zhu, H., Dick, J.: Discrepancy bounds for deterministic acceptance-rejection samplers. Electr. J. Stat. 8(1), 678–707 (2014). https://doi.org/10.1214/14-EJS898
Acknowledgements
We thank the reviewers for their comments, which helped us improve this chapter. The second and third authors are grateful for the financial support of NSERC via grant RGP 238959 and RGPIN-2020-04897, respectively.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hintz, E., Hofert, M., Lemieux, C. (2022). Quasi-Random Sampling with Black Box or Acceptance-Rejection Inputs. In: Botev, Z., Keller, A., Lemieux, C., Tuffin, B. (eds) Advances in Modeling and Simulation. Springer, Cham. https://doi.org/10.1007/978-3-031-10193-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-031-10193-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-10192-2
Online ISBN: 978-3-031-10193-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)