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Biometrika 1967 54(1-2):293-300; doi:10.1093/biomet/54.1-2.293
© 1967 by Biometrika Trust
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Reduction of the multivariate normal integral to characteristic form

DONALD R. CHILDS *

Laboratory for Electronics Boston, Massachusetts

The nth order integral over a generalized quadrant of the multivariate normal distribution has first been transformed to a series of integrals over the entire hypervolume for all orders. The series has been derived for all n and is exhibited explicitly for all orders up to n = 6. The fourth-order integral has further been reduced to three first-order integrals, thus making them amenable to machine solution. This method avoids the difficulty of trying to integrate a mixed distribution over an interval when the variables cannot be separated. Three very special cases of the sixth-order integral are exhibited. It can be shown that the sixth-order integral can be reduced to five second-order integrals.


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