On a lower bound for the multivariate normal Mills' ratio
Let \(x=(x_ 1,x_ 2,...,x_ n)'\) be a standardized multivariate normal vector with \(Ex_ ix_ j=\phi_{ij}\) and \(\Sigma =(\phi_{ij})\). The multivariate Mills' ratio is defined as \[ R(a;\Sigma)=P(x\geq a)/\Phi (a;\Sigma) \] where \(x\geq a\) means \(x_ i\geq a_ i\) for all i, and \(\Phi\) is the density of x. Several approximations for R(a;\(\Sigma)\) have been proposed [see \textit{G. P. Steck}, ibid. 7, 547-551 (1979; Zbl 0399.62047)]. Steck also proposed three more approximations using a likelihood ratio argument. The lower bound \(\hat R_ 2\) is the best of the three bounds. The author has obtained a certain class of densities with positive skewness parameter. Using this result and under some conditions on the correlation matrix \(\Sigma\), he has obtained Steck's best approximation \(\hat R_ 2\) to be a lower bound for R(a;\(\Sigma)\). Further the author has conjectured that there are correlation matrices \(\Sigma\) which do not satisfy the hypotheses but for which \(\hat R_ 2\) is still a lower bound.
scientific article; zbMATH DE number 3980123