On the computation of a bivariate $t$-distribution
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- by D. E. Amos and W. G. Bulgren PDF
- Math. Comp. 23 (1969), 319-333 Request permission
Abstract:
The cumulative bivariate $t$-distribution associated with random variables ${T_1} = {X_1}/{(S/k)^{1/2}}$, ${T_2} = {X_2}/{(S/k)^{1/2}}$ is considered where ${X_1}$, ${X_2}$ are bivariate normal with correlation coefficient $\rho$ and $S$ is an independent ${\chi ^2}$ random variable with $k$ degrees of freedom. Representations in terms of series and simple, one-dimensional quadratures are presented together with efficient computational procedures for the special functions used in numerical evaluation.References
-
C. W. Dunnett, “A multiple comparison procedure for comparing several treatments with a control,” J. Amer. Statist. Assoc., v. 50, 1955, pp. 1096–1121.
- Charles W. Dunnett and Milton Sobel, A bivariate generalization of Student’s $t$-distribution, with tables for certain special cases, Biometrika 41 (1954), 153–169. MR 61793, DOI 10.1093/biomet/41.1-2.153
- C. W. Dunnett and M. Sobel, Approximations to the probability integral and certain percentage points of a multivariate analogue of Student’s $t$-distribution, Biometrika 42 (1955), 258–260. MR 68169, DOI 10.1093/biomet/42.1-2.258 A. Erdélyi, et al., Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953. MR 15, 419. A. Erdélyi, et al., Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. MR 15, 419. A. Erdélyi, et al., Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954. MR 15, 868.
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
- Shanti S. Gupta and Milton Sobel, On a statistic which arises in selection and ranking problems, Ann. Math. Statist. 28 (1957), 957–967. MR 93846, DOI 10.1214/aoms/1177706796 S. S. Gupta, “Probability integrals of multivariate normal and multivariate $t$,” Ann. Math. Statist., v. 34, 1963, pp. 792–828. MR 27 #2048.
- S. John, On the evaluation of the probability integral of the multivariate $t$-distribution, Biometrika 48 (1961), 409–417. MR 144406, DOI 10.1093/biomet/48.3-4.409
- S. John, Methods for the evaluation of probabilities of polygonal and angular regions when the distribution is bivariate $t$, Sankhyā Ser. A 26 (1964), 47–54. MR 198581 P. R. Krishnaiah & J. V. Armitage, Percentage Points of the Multivariate $t$-Distribution, ARL 65–199, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio, 1965.
- N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795, DOI 10.1063/1.3047047
- D. B. Owen, A special case of a bivariate non-central $t$-distribution, Biometrika 52 (1965), 437–446. MR 205361, DOI 10.1093/biomet/52.3-4.437
- K. C. S. Pillai and K. V. Ramachandran, On the distribution of the ratio of the $i$th observation in an ordered sample from a normal population to an independent estimate of the standard deviation, Ann. Math. Statistics 25 (1954), 565–572. MR 64356, DOI 10.1214/aoms/1177728724
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 319-333
- MSC: Primary 65.25; Secondary 62.00
- DOI: https://doi.org/10.1090/S0025-5718-1969-0242348-0
- MathSciNet review: 0242348