STOCHASTIC ORDERING OF MULTIVARIATE NORMAL DISTRIBUTIONS

ALFRED MÜLLER

Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128 Karlsruhe, Germany, email: mueller@wior.uni-karlsruhe.de

(Received December 6, 1999)

Abstract. We show an interesting identity for Ef(Y Y Y Y Y Y Y Y) − Ef(X X X X X X X X), where X X X X X X X X, Y Y Y Y Y Y Y Y are normally distributed random vectors and f is a function fulfilling some weak regularity condition. This identity will be used for a unified derivation of sufficient conditions for stochastic ordering results of multivariate normal distributions, some well known ones as well as some new ones. Moreover, we will show that many of these conditions are also necessary. As examples we will consider the usual stochastic order, convex order, upper orthant order, supermodular order and directionally convex order.

Key words and phrases: Multivariate normal distribution, stochastic orders, supermodular order, directionally convex order.