Bivariate conditioning approximations for multivariate normal probabilities

New formulas are derived for multivariate normal probabilities defined for hyper-rectangular probability regions. The formulas use conditioning with a sequence of bivariate normal probabilities. The result is an approximate formula for multivariate normal probabilities which uses a product of bivariate normal probabilities. The new approximation method is compared with approximation methods based on products of univariate normal probabilities, using tests with random covariance-matrix/probability-region problems for up to twenty variables. The reordering of variables is studied to improve efficiency of the new method.

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