Evaluation of Normal Probabilities of Symmetric Regions

Let ${\bf X} = (X_1 , \cdots ,X_n )'$ be a standardized multivariate normal random vector and let $A \in \mathbb{R}^n $ be a permutation-symmetric region. We provide and justify an approximation to $P({\bf X} \in A)$ which is easy to compute. We then consider the special case of exceedance probabilities, ${\bf P}$ (k of the n components of ${\bf X}$ exceed a constant c), and use it to show how our approximation can be used to improve upon procedures for constructing confidence sets for the mean of a multivariate normal. We also assess the performance of our approximation by using numerical examples.

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