An asymptotic expansion for the multivariate normal distribution and Mills' ratio

An asymptotic expansion for the multivariate normal integral over an infinitely ex tended rectangle, and therefore also for the associated multivariate Mills ' ratio , is developed. The expansion is valid provided the vertex of the rectangle lies in a polyhedral half·cone determined by the set of regress ion planes. The expansion obtained here is a natural generalization of the c lass ic ex pansion for the normal univariate integr al, and the coeffi cients in it involve the mome nts of the conjugate multinormal di stribution.

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