Numerical computation of rectangular bivariate and trivariate normal and t probabilities

Statistics and Computing

Volumn 14, Issue 3, 2004, Pages 251-260

  Genz, Alan ^a  

^a WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

bivariate normal; bivariate t; distribution; Plackett formula; trivariate normal; trivariate t

Indexed keywords

EID: 3843147201     PISSN: 09603174     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:STCO.0000035304.20635.31     Document Type: Review

References (26)

1. Andel J. 1974. On evaluation of some two-dimensional normal probabilities. Aplikace Matematiky 19: 28-35.
2. Cornish E.A. 1954. The multivariate t-distribution associated with a set of normal sample deviates. Australian Journal of Physics 7: 531-542.
3. Cox D.R. and Wermuth N. 1991. A simple approximation for bivariate and trivariate normal integrals. International Statistics Review 59: 263-269.
4. Davis P.J. and Rabinowitz P. 1984. Methods of Numerical Integration. Academic Press, New York.
5. Drezner Z. 1978. Computation of the bivariate normal integral. Mathematics of Computation 32: 277-279.
6. Drezner Z. and Wesolowsky G.O. 1989. On the computation of the bivariate normal integral. Journal of Statist. Comput. Simul. 35: 101-107.
7. Drezner Z. 1992. Computation of the multivariate normal integral. ACM Transactions on Mathematics Software 18: 450-460.
8. Drezner Z. 1994. Computation of the trivariate normal integral. Mathematics of Computation 62: 289-294.
9. Donnelly TG. 1963. Algorithm 462: Bivariate normal distribution. Communications of the ACM 16: 638.
10. Dunnett C.W. and Sobel M. 1954. A bivariate generalization of student's t-distribution, with tables for certain special cases. Biometrika 41: 153-169.
11. Gassmann H.I. 2000. Rectangle probabilities of trivariate normal distributions, technical report available at http://www.mgmt.dal.ca/sba/profs/ hgassmann/.
12. Gassmann H.I., Deák I., and Szántai T. 2002. Computing multivariate normal probabilities: A new look. Journal of Computational and Graphical Statistics 11: 920-949.
13. Genz A. 1993. Comparison of methods for the computation of multivariate normal probabilities. Computing Science and Statistics 25: 400-405.
14. Genz A. and Bretz F. 2002. Comparison of methods for the computation of multivariate t probabilities. J. Comp. Graph. Stat. 11: 950-971.
15. Genz A., Bretz F., and Hochberg Y. 2003. Approximations to multivariate t probabilities with application to multiple comparison procedures, technical report available at http://www.math.wsu.edu/faculty/genz.
16. Mee R.W. and Owen D.B. 1983. A simple approximation for bivariate normal probability. Journal of Quality Technology 15: 72-75.
17. Owen D.B. 1956. Tables for computing bivariate normal probability. Annals of Mathematical Statistics 27: 1075-1090.
18. Patefield M. and Tandy D. 2000. Fast and accurate computation of Owen's T-function. Journal of Statistical Software 5(5): http://www.jstatsoft.org.
19. Plackett R.L. 1954. A reduction formula for normal multivariate probabilities. Biometrika 41: 351-360.
20. Piessens R., deDoncker E., Uberhuber C., and Kahaner D. 1983. QUADPACK: A Subroutine Package for Automatic Integration. Springer-Verlag, New York.
21. Schervish M. 1984. Multivariate normal probabilities with error bound. Applied Statistics 33: 81-87.
22. Sheppard W.F. 1900. On the computation of the double integral expressing normal correlation. Transactions of the Cambridge Philosophical Society 19:23-69.
23. Steen N.M., Byrne G.O., and Gelhard E.M. 1969. Gaussian quadratures. Mathematics of Computation 23: 661-671.
24. Terza J.V. and Welland U. 1988. A comparison of bivariate normal algorithms. Journal of Statist. Comput. Simul. 19: 115-127.
25. Tong Y.L. 1990. The Multivariate Normal Distribution. Springer-Verlag, New York.
26. Wang M. and Kennedy W.J. 1992. A numerical method for accurately approximating multivariate normal probability. Computational Statistics and Data Analysis 13: 197-210.