On computation of the bivariate normal distribution
Author:
D. E. Amos
Journal:
Math. Comp. 23 (1969), 655659
MSC:
Primary 65.25; Secondary 62.00
MathSciNet review:
0247733
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Abstract 
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Abstract: A quadrature and two series representations are given as limiting cases of a bivariate distribution. The quadrature is taken over the complementary error function and the series are sums of Bessel functions and incomplete beta functions, respectively. Comparisons with some known results are made in terms of accuracy and computer time.
 [1]
Milton
Abramowitz and Irene
A. Stegun, Handbook of mathematical functions with formulas,
graphs, and mathematical tables, National Bureau of Standards Applied
Mathematics Series, vol. 55, For sale by the Superintendent of
Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
(29 #4914)
 [2]
D.
E. Amos and W.
G. Bulgren, On the computation of a bivariate
𝑡distribution, Math. Comp. 23 (1969), 319–333. MR 0242348
(39 #3679), http://dx.doi.org/10.1090/S00255718196902423480
 [3]
C. W. Clenshaw, Chebyshev Series for Mathematical Functions, National Physical Lab. Math. Tables, vol. 5, HMSO, London, 1962. MR 26 #362.
 [4]
W. Gautschi, "Algorithm 222incomplete beta function ratios," Comm. Assoc. Comput. Mach., v. 7, 1964, pp. 143144; "Certification of algorithm 222," Ibid., v. 244, 1964.
 [5]
W. Gautschi, "Algorithm 236Bessel functions of the first kind," Comm. Assoc. Comput. Mach., v. 7, 1964, pp. 479480; "Certification of algorithm 236," Ibid., v. 8, 1965, pp. 105106.
 [6]
Walter
Gautschi, Computational aspects of threeterm recurrence
relations, SIAM Rev. 9 (1967), 24–82. MR 0213062
(35 #3927)
 [7]
Shanti
S. Gupta, Bibliography on the multivariate normal integrals and
related topics, Ann. Math. Statist. 34 (1963),
829–838. MR 0152069
(27 #2049)
 [8]
Y.
L. Luke and J.
Wimp, Jacobi polynomial expansions of a
generalized hypergeometric function over a semiinfinite ray, Math. Comp. 17 (1963), 395–404. MR 0157014
(28 #255), http://dx.doi.org/10.1090/S00255718196301570144
 [9]
Donald
B. Owen, Tables for computing bivariate normal probabilities,
Ann. Math. Statist. 27 (1956), 1075–1090. MR 0127562
(23 #B607)
 [10]
W. F. Sheppard, "On the calculation of the double integral expressing normal correlation," Trans. Cambridge Philos. Soc., v. 19, 1900, pp. 2369.
 [11]
Tables of the Bivariate Normal Distribution Function and Related Functions, Nat. Bur. Standards Appl. Math. Series, 50, Superintendent of Documents, U. S. Government Printing Office, 1959.
 [1]
 M. Abramowitz & I. A. Stegun, (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing, with corrections, 1965. MR 29 #4914; MR 31 #1400. MR 0167642 (29:4914)
 [2]
 D. E. Amos & W. G. Bulgren, "On the computation of a bivariate distribution," Math. Comp., v. 23, 1969, pp. 319333. MR 0242348 (39:3679)
 [3]
 C. W. Clenshaw, Chebyshev Series for Mathematical Functions, National Physical Lab. Math. Tables, vol. 5, HMSO, London, 1962. MR 26 #362.
 [4]
 W. Gautschi, "Algorithm 222incomplete beta function ratios," Comm. Assoc. Comput. Mach., v. 7, 1964, pp. 143144; "Certification of algorithm 222," Ibid., v. 244, 1964.
 [5]
 W. Gautschi, "Algorithm 236Bessel functions of the first kind," Comm. Assoc. Comput. Mach., v. 7, 1964, pp. 479480; "Certification of algorithm 236," Ibid., v. 8, 1965, pp. 105106.
 [6]
 W. Gautschi, "Computational aspects of threeterm recurrence relations," SIAM Rev., v. 9, 1967, pp. 2482. MR 0213062 (35:3927)
 [7]
 S. S. Gupta, "Bibliography on the multivariate normal integrals and related topics," Ann. Math, Statist., v. 34, 1963, pp. 829838. MR 27 #2049. MR 0152069 (27:2049)
 [8]
 Y. L. Luke & J. Wimp, "Jacobi polynomial expansions of a generalized hypergeometric function over a semiinfinite ray," Math. Comp., v. 17, 1963, pp. 395404. MR 28 #255. MR 0157014 (28:255)
 [9]
 D. B. Owen, "Tables for computing bivariate normal probabilities," Ann. Math. Statist., v. 27, 1956, pp. 10751090. MR 0127562 (23:B607)
 [10]
 W. F. Sheppard, "On the calculation of the double integral expressing normal correlation," Trans. Cambridge Philos. Soc., v. 19, 1900, pp. 2369.
 [11]
 Tables of the Bivariate Normal Distribution Function and Related Functions, Nat. Bur. Standards Appl. Math. Series, 50, Superintendent of Documents, U. S. Government Printing Office, 1959.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718196902477339
PII:
S 00255718(1969)02477339
Article copyright:
© Copyright 1969 American Mathematical Society
