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An asymptotic expansion for a class of multivariate normal integrals

Published online by Cambridge University Press:  09 April 2009

Harold Ruben
Harold Ruben
Affiliation:
Sheffield University
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Let x = (x1, x2,…, xn) be a normal random vector with zero expectation vector and with a variance-covariance matrix which has 1 for its diagonal elements and ρ for its off-diagonal elements. Consider the quantity where.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 3 , February 1962 , pp. 253 - 264
Copyright
Copyright © Australian Mathematical Society 1962

References

[1]Godwin, H. J., “On the distribution of the estimate of the mean deviation obtained from samples from a normal population,” Biometrika, Vol. 33 (1945), pp. 254–256.CrossRefGoogle ScholarPubMed
[2]Godwin, H. J., “A further note on the mean deviation”, Biometrika, Vol. 35 (1948), pp. 304–309.CrossRefGoogle Scholar
[3]Kudô, A., “On the distribution of the maximum value of an equally correlated sample from a normal population”, Sankhyā, Vol. 20 (1958), pp. 309–316.Google Scholar
[4]Nair, K. R., “The distribution of the extreme deviate from the sample mean and its studentised form”, Biometrika, Vol. 35 (1948), pp. 118–144.CrossRefGoogle Scholar
[5]Harold, Ruben, “Probability content of regions under spherical normal distributions, I,” Ann. Math. Stat., Vol. 31 (1960), pp. 598–618.Google Scholar
[6]Harold, Ruben, “Probability content of regions under spherical normal distributions, III: The bivariate normal integral,” Ann. Math. Stat., Vol. 32 (1961), pp. 171–186.Google Scholar
[7]Harold, Ruben, “A power series expansion for a çlass of Schläfli functions,” J. London Math. Soc., Vol. 36 (1961), pp. 69–77.Google Scholar
[8]Harold, Ruben, “On the numerical evaluation of a class of multivariate normal integrals,” Proc. Roy. soc. Edin., Vol. 65, (1961), pp. 272–281.Google Scholar
[9]Harold, Ruben, “On the moments of order statistics in samples from normal populations,” Biometrika, Vol. 41 (1954), pp. 200–227.Google Scholar
[10]Watson, G. N., Bessel Functions. Cambridge University Press, 1922.Google Scholar