Simulation from the Tail of the Univariate and Multivariate Normal Distribution

  • Zdravko BotevEmail author
  • Pierre L’Ecuyer
Part of the EAI/Springer Innovations in Communication and Computing book series (EAISICC)


We study and compare various methods to generate a random variate or vector from the univariate or multivariate normal distribution truncated to some finite or semi-infinite region, with special attention to the situation where the regions are far in the tail. This is required in particular for certain applications in Bayesian statistics, such as to perform exact posterior simulations for parameter inference, but could have many other applications as well. We distinguish the case in which inversion is warranted, and that in which rejection methods are preferred.


  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970)zbMATHGoogle Scholar
  2. 2.
    J.M. Blair, C.A. Edwards, J.H. Johnson, Rational Chebyshev approximations for the inverse of the error function. Math. Comput. 30, 827–830 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Z.I. Botev, The normal law under linear restrictions: simulation and estimation via minimax tilting. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 79(1), 125–148 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Z.I. Botev, P. L’Ecuyer, Efficient estimation and simulation of the truncated multivariate Student-t distribution, in Proceedings of the 2015 Winter Simulation Conference (IEEE Press, Piscataway, 2015), pp. 380–391Google Scholar
  5. 5.
    Z.I. Botev, P. L’Ecuyer, Simulation from the normal distribution truncated to an interval in the tail, in 10th EAI International Conference on Performance Evaluation Methodologies and Tools, 25th–28th October 2016 Taormina (ACM, New York, 2017), pp. 23–29Google Scholar
  6. 6.
    Z.I. Botev, M. Mandjes, A. Ridder, Tail distribution of the maximum of correlated Gaussian random variables, in Proceedings of the 2015 Winter Simulation Conference (IEEE Press, Piscataway, 2015), pp. 633–642Google Scholar
  7. 7.
    N. Chopin, Fast simulation of truncated Gaussian distributions. Stat. Comput. 21(2), 275–288 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. Devroye, Non-Uniform Random Variate Generation (Springer, New York, NY, 1986)CrossRefGoogle Scholar
  9. 9.
    J. Geweke, Efficient simulation of the multivariate normal and Student-t distributions subject to linear constraints and the evaluation of constraint probabilities, in Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Fairfax, VA, 1991, pp. 571–578Google Scholar
  10. 10.
    E. Hashorva, J. Hüsler, On multivariate Gaussian tails. Ann. Inst. Stat. Math. 55(3), 507–522 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    W. Hörmann, J. Leydold, G. Derflinger, Automatic Nonuniform Random Variate Generation (Springer, Berlin, 2004)CrossRefGoogle Scholar
  12. 12.
    D.P. Kroese, T. Taimre, Z.I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011)CrossRefGoogle Scholar
  13. 13.
    P. L’Ecuyer, Variance reduction’s greatest hits, in Proceedings of the 2007 European Simulation and Modeling Conference, Ghent (EUROSIS, Hasselt, 2007), pp. 5–12Google Scholar
  14. 14.
    P. L’Ecuyer, Quasi-Monte Carlo methods with applications in finance. Finance Stochast. 13(3), 307–349 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    P. L’Ecuyer, Random number generation with multiple streams for sequential and parallel computers, in Proceedings of the 2015 Winter Simulation Conference, pp. 31–44 (IEEE Press, New York, 2015)Google Scholar
  16. 16.
    P. L’Ecuyer, SSJ: stochastic simulation in Java, software library (2016).
  17. 17.
    J. Leydold, UNU.RAN—Universal Non-Uniform RANdom number generators (2009). Available at
  18. 18.
    G. Marsaglia, Generating a variable from the tail of the normal distribution. Technometrics 6(1), 101–102 (1964)Google Scholar
  19. 19.
    G. Marsaglia, T.A. Bray, A convenient method for generating normal variables. SIAM Rev. 6, 260–264 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    G. Marsaglia, A. Zaman, J.C.W. Marsaglia, Rapid evaluation of the inverse normal distribution function. Stat. Probab. Lett. 19, 259–266 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.P. Mills, Table of the ratio: area to bounding ordinate, for any portion of normal curve. Biometrika 18(3/4), 395–400 (1926)CrossRefGoogle Scholar
  22. 22.
    C.P. Robert, Simulation of truncated normal variables. Stat. Comput. 5(2), 121–125 (1995)CrossRefGoogle Scholar
  23. 23.
    R.I. Savage, Mills’ ratio for multivariate normal distributions. J. Res. Nat. Bur. Standards Sect. B 66, 93–96 (1962)CrossRefGoogle Scholar
  24. 24.
    D.B. Thomas, W. Luk, P.H. Leong, J.D. Villasenor, Gaussian random number generators. ACM Comput. Surv. 39(4), Article 11 (2007)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.UNSW SydneySydneyAustralia
  2. 2.Université de MontréalMontréalCanada
  3. 3.Inria - RennesRennesFrance

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