Using Complex Integration to Compute Multivariate Normal Probabilities

References

• Abramowitz , M. and Stegun , I. A. 1972 . Handbook of Mathematical Functions , New York : Dover Publications Inc. .
   Google Scholar
• Bland , R. P. and Owen , D. B. 1966 . “A Note on Singular Normal Distributions” . Annals of the Institute of Statistical Mathematics , 18 : 113 – 116 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Chung , K. L. 1974 . A Course in Probability Theory, , 2nd ed. , New York : Academic Press .
   Google Scholar
• Davis , P. J. and Rabinowitz , P. 1984 . Methods of Numerical Integration, , 2nd ed. , New York : Academic Press .
   Google Scholar
• Dunnett , C. W. 1955 . “A Multiple Comparison Procedure for Comparing Several Treatments with a Control” . Journal of the American Statistical Association , 50 : 1096 – 1121 .
   Web of Science ®Google Scholar
• Dunnett , C. W. 1989 . “Multivariate Normal Probability Integrals with Product Correlation Structure” . Applied Statistics , 38 : 564 – 579 .
   Web of Science ®Google Scholar
• Dunnett , C. W. 1993 . “Correction to Algorithm as 251: Multivariate Normal Probability Integrals with Product Correlation Structure” . Applied Statistics , 42 : 709
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Fritsch , K. and Hsu , J. C. 1997 . “On Analysis of Means” . In Advances in Statistical Decision Theory and Methodology , Edited by: Balakrishnan , N. and Panchapakesan , S. Boston : Birkhauser .
   Google Scholar
• Gautschi , W. 1969 . “Algorithm 363, Complex Error Function” . Communications of the ACM , 12 : 635
   Web of Science ®Google Scholar
• Gautschi , W. 1970 . “Efficient Computation of the Complex Error Function” . SIAM Journal of Mathematical Analysis , 7 : 187 – 198 .
   Google Scholar
• Goodwin , E. T. and Staton , J. 1948 . “Table of du” . The Quarterly Journal of Mechanics and Applied Mathematics , 1 : 319 – 326 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Hochberg , Y. and Tamhane , A. C. 1987 . Multiple Comparison Procedures , New York : John Wiley .
   Google Scholar
• Hsu , J. C. 1996 . Multiple Comparisons: Theory and Methods , London : Chapman & Hall .
   Google Scholar
• IMSL . 1995a . MATH/LIBRARY Special Functions Version 3.0 , Houston , TX : Visual Numerics .
   Google Scholar
• IMSL . 1995b . MATH/LIBRARY Version 3.0 , Houston , TX : Visual Numerics .
   Google Scholar
• Kramer , C. Y. 1956 . “Extension of Multiple Range Tests to Group Means with Unequal Numbers of Replications” . Biometrics , 12 : 309 – 310 .
   Web of Science ®Google Scholar
• Nelson , P. R. 1982 . “Multivariate Normal and t Distributions with ρij = αiαk” . Communications in Statistics–Simulation and Computation , 11 : 239 – 248 .
   Web of Science ®Google Scholar
• Nelson , P. R. “Numerical Evaluation of Multivariate Normal Integrals with Correlations ρij = αiαj” . The Frontiers of Statistical Scientific Theory & Industrial Applications (Proceedings of ICOSCO-I, The First International Conference on Statistical Computing) . vol. II , pp. 97 – 113 .
   Google Scholar
• Nelson , P. R. 1993 . “Additional Uses for the Analysis of Means and Extended Tables of Critical Values” . Technometrics , 35 : 61 – 71 .
   Web of Science ®Google Scholar
• 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 .
   Google Scholar
• Šidák , Z. 1967 . “Rectangular Confidence Regions for the Means of Multivariate Normal Distributions” . Journal of the American Statistical Association , 62 : 626 – 633 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Steck , G. P. and Owen , D. B. 1962 . “A Note on the Equi-correlated Multivariate Normal Distribution” . Biometrika , 49 : 269 – 271 .
   Web of Science ®Google Scholar
• Stoer , J. and Bulirsch , J. 1967 . “Handbook Series Numerical Integration, Numerical Quadrature by Extrapolation” . Numerische Mathematik , 9 : 271 – 278 .
   Web of Science ®Google Scholar
• Stoer , J. and Bulirsch , J. 1980 . Introduction to Numerical Analysis , New York : Springer-Verlag .
   Google Scholar
• Tukey , J. W. 1953 . “The Problem of Multiple Comparisons” , Department of Statistics, Princeton University . dittoed manuscript of 396 pages
   Google Scholar
• 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 .
   Web of Science ®Google Scholar