The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if ;[varrho]ij; < 1/(k - 1), 1 <= i < j <= k, where [varrho]ij are the correlation coefficients of a k-variate normal distribution. The tetrachoric series for orthant probabilities diverges whenever k is even and [varrho]ij > 1/(k - 1) or k is odd and [varrho]ij > 1/(k - 2), 1 <= i < j <= k. Other specific results concerning the convergence or divergence of this series are also given. The principal point is that the assertion that the tetrachoric series converges for all k >= 2 and all [varrho]ij such that the correlation matrix is positive definite is false.