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Probability integrals of multivariate normal and multivariate /. Ann. Math. Statist. 34, 792-828. Hardy, G. , Littlewood, J. , and Polya, G. (1959). Inequalities, 2nd ed. Cambridge Univ. Press, London and New York. Jogdeo, K. (1970). A simple proof of an inequality for multivariate normal probabilities of rectangles. Ann. Math. Statist. 41, 1357-1359. Khatri, C. G. (1967). On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist.

Remarks. , for every fixed λ [ J A+\y f(x)dx=[ J A-\y f(x)dx. 1 can be restated as the following: Under the conditions given the integral fA+Xyf(x)dx is monotonically nonincreasing in |λ| for every fixed y. 2)) becomes an equality if and only if [(A+y)nDu]=[(AnDu) + y] holds for every u > 0. If / is a normal density with mean vector 0 and a positive definite covariance matrix Σ, then / is unimodal and this condition is not satisfied (Anderson, 1955). Hence for normal distribution the inequality is strict.

It is clear that if / ( x ) (or A) is sign invariant, then it is symmetric about the origin, but not conversely. 5) is also sign invariant. With this stronger symmetry property we can now prove the following theorem. 58 4. 4. Let /(x): <3l*->[0, oo) bè sign invariant and unimodal. Let Actflk be sign invariant and convex. 16) A+\y\ and the inequality ^A+y* > ( J A+y holds for all y,y* satisfying |y| > |y*|. 5. 1. Note that in this proof the Brunn-Minkowski theorem is not needed. Proof. , Xj _ | , Xj, Xj + j , .