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XX.—On the Numerical Evaluation of a Class of Multivariate Normal Integrals*

Published online by Cambridge University Press:  14 February 2012

Harold Ruben
Harold Ruben
Affiliation:
Columbia University, N.Y., U.S.A.
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Synopsis

The probability that each of n equally correlated normal random variables shall not fall short of a given value h is obtained as the product of the joint density function of the variables at the cut-off point (h, h,…, h) and an infinite power series in h. The coefficients in the latter series may be interpreted geometrically as the moments of a regular (n – l)-dimensional spherical simplex with common dihedral angles arc cos –p relative to a certain plane of symmetry. These moments may in their turn be expressed as linear functions of the measures of regular hyperspherical simplices of various dimensionalities, tables of which are available elsewhere.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 65 , Issue 3 , 1960 , pp. 272 - 281
Copyright
Copyright © Royal Society of Edinburgh 1960

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References

REFERENCES TO LITERATURE

[1] Cramer, H., 1946. Mathematical Methods of Statistics (Princeton University Press).Google Scholar
[2] Das, S. C., 1956. “The Numerical Evaluation of a Class of Integrals, II”, Proc. Camb. Phil. Soc., 52, 442–448.CrossRefGoogle Scholar
[3] Dunnett, C. W., 1955. “Statistical Need for New Tables Involving the Multivariate Normal Distribution”, (presented at the 18th Annual Meeting of the Institute of Mathematical Statistics, New York).Google Scholar
[4] 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.CrossRefGoogle Scholar
[5] Kendall, M. G., 1954. Contribution to discussion of paper “Distributionfree Rests Based on Records” by Foster, F. G., and Stuart, A., J. Roy. Statist. Soc., B, 16, 1–22.Google Scholar
[6] Moran, P. A. P., 1956. “The Numerical Evaluation of a Class of Integrals”, Proc. Camb. Phil. Soc., 52, 230–233.CrossRefGoogle Scholar
[7] National Bureau of Standards. Unpublished tables.Google Scholar
[8] Ruben, H., 1954. “On the Moments of Order Statistics in Samples from Normal Populations”, Biometrika, 41, 201–227.CrossRefGoogle Scholar
[9] Ruben, H., 1960. “On the Geometrical Moments of Skew-regular Simplices in Hyperspherical Space, with some Applications in Geometry and Mathematical Statistics”, Ada Math. Stockh., 103, 1–23.Google Scholar
[10] Ruben, H., 1961. “Probability Content of Regions under Spherical Normal Distributions, III: The Bivariate Normal Integral”, Ann. Math. Statist., 32, 171–186.CrossRefGoogle Scholar
[11] Stuart, A., 1958. “Equally Correlated Variates and the Multinormal Integral” J. Roy. Statist. Soc., B, 20, 373–378.Google Scholar
[12] Thomson, G. H., 1939. The Factorial Analysis of Human Ability, (London).CrossRefGoogle Scholar