Using complex integration to compute multivariate normal probabilities

Journal of Computational and Graphical Statistics

Volumn 6, Issue 4, 1997, Pages 397-415

  Soong, W C ^a   Hsu, Jason C ^b  

^a NATIONAL CHUNG HSING UNIVERSITY   (Taiwan)

^b OHIO STATE UNIVERSITY   (United States)

Author keywords

Complex integration; Multiple comparisons; Multivariate normal; Rectangular probability

Indexed keywords

EID: 0031320377     PISSN: 10618600     EISSN: 15372715     Source Type: Journal    
DOI: 10.1080/10618600.1997.10474750     Document Type: Article

References (31)

1. Abramowitz, M., and Stegun, I. A. (eds) (1972), Handbook of Mathematical Functions, New York: Dover Publications Inc.
2. Bland, R. P., and Owen, D. B. (1966), “A Note on Singular Normal Distributions,” Annals of the Institute of Statistical Mathematics, 18, 113-116.
3. Campbell, P. F., and McCabe, G. P. (1984), “Predicting the Success of Freshmen in a Computer Science Major,” Communications of the ACM, 27, 1108-1113.
4. Chung, K. L. (1974), A Course in Probability Theory (2nd ed.), New York: Academic Press.
5. Davis, P. J., and Rabinowitz, P. (1984), Methods of Numerical Integration (2nd ed.), New York: Academic Press.
6. Dunnett, C. W. (1955), “A Multiple Comparison Procedure for Comparing Several Treatments with a Control,” Journal of the American Statistical Association, 50, 1096-1121.
7. Dunnett, C. W. (1989), “Multivariate Normal Probability Integrals with Product Correlation Structure,” Applied Statistics, 38, 564-579.
8. Dunnett, C. W. (1993), “Correction to Algorithm as 251: Multivariate Normal Probability Integrals with Product Correlation Structure,” Applied Statistics, 42, 709.
9. Dunnett, C. W„ and Sobel, M. (1955), “Approximations to the Probability Integral and Certain Percentage Points of a Multivariate Analogue of Student’s t-Distribution,” Biometrika, 42, 258-260.
10. Fritsch, K., and Hsu, J. C. (1997), “On Analysis of Means,” in Advances in Statistical Decision Theory and Methodology, eds. N. Balakrishnan and S. Panchapakesan, Boston: Birkhauser.
11. Gautschi, W. (1969), “Algorithm 363, Complex Error Function,” Communications of the ACM, 12, 635.
12. Gautschi, W. (1970), “Efficient Computation of the Complex Error Function,” SIAM Journal of Mathematical Analysis, 1, 187-198.
13. Goodwin, E. T., and Staton, J. (1948), “Table of, The Quarterly Journal of Mechanics and Applied Mathematics, 1, 319-326.
14. Halperin, M., Greenhouse, S. W., Cornfield, J., and Zalokar, J. (1955), “Tables of Percentage Points for the Studentized Maximum Absolute Deviate in Normal Samples,” Journal of the American Statistical Association, 50, 185-195.
15. Hayter, A. J. (1984), “A Proof of the Conjecture that the Tukey-Kramer Multiple Comparisons Procedure is Conservative,” The Annals of Statistics, 12,61-75.
16. Hochberg, Y., and Tamhane, A. C. (1987), Multiple Comparison Procedures, New York: John Wiley.
17. Hsu, J. C. (1996), Multiple Comparisons: Theory and Methods, London: Chapman & Hall.
18. IMSL (1995a), MATH/LIBRARY Special Functions Version 3.0, Houston, TX: Visual Numerics.
19. IMSL (1995b). MATH/LIBRARY Version 3.0, Houston, TX: Visual Numerics.
20. Kramer, C. Y. (1956), “Extension of Multiple Range Tests to Group Means with Unequal Numbers of Replications,” Biometrics, 12, 309-310.
21. Nelson, P. R. (1982), “Multivariate Normal and t Distributions with pij = onafi Communications in Statistics-Simulation and Computation, 11, 239-248.
22. Nelson, P. R. (1991), “Numerical Evaluation of Multivariate Normal Integrals with Correlations,” in The Frontiers of Statistical Scientific Theory & Industrial Applications (Proceedings of ICOSCO-I, The First International Conference on Statistical Computing), vol. II, pp. 97-113.
23. Nelson, P. R. (1993), “Additional Uses for the Analysis of Means and Extended Tables of Critical Values,” Technometrics, 35, 61-71.
24. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1986), Numerical Recipes: The Art of Scientific Computing, New York: Cambridge University Press.
25. Sidék, Z. (1967), “Rectangular Confidence Regions for the Means of Multivariate Normal Distributions,” Journal of the American Statistical Association, 62, 626-633.
26. Spurrier, J. D., and Isham, S. P. (1985), “Exact Simultaneous Confidence Intervals of Pairwise Comparisons of Three Normal Means,” Journal of the American Statistical Association, 80, 438-442.
27. Steck, G. P., and Owen, D. B. (1962), “A Note on the Equi-correlated Multivariate Normal Distribution,” Biometrika, 49, 269-271.
28. Stoer, J., and Bulirsch, J. (1967), “Handbook Series Numerical Integration, Numerical Quadrature by Extrapolation,” Numerische Mathematik, 9, 271-278.
29. Stoer, J., and Bulirsch, J. (1980), Introduction to Numerical Analysis, New York: Springer-Verlag.
30. Tukey, J. W. (1953), “The Problem of Multiple Comparisons,” dittoed manuscript of 396 pages, Department of Statistics, Princeton University.
31. Uusipaikka, E. (1985), “Exact Simultaneous Confidence Intervals for Multiple Comparisons of Three or Four Mean Values,” Journal of the American Statistical Association, 80, 196-201.