Annals of Operations Research 32(1991)127-140 127 CONDITIONAL VARIABILITY ORDERING OF DISTRIBUTIONS C. METZGER and L. ROSCHENDORF Institute of Mathematical Statistics, University of Miinster, D-4400 Miinster, Germany Abstract The main idea of conditional ordering is to sharpen the ordering results in such a way that even conditionally on some kind of additional information on the underlying "experiment" the ordering is valid. In this paper, some sufficient conditions for conditional variability and conditional dispersion orderings are established. The main idea is to find sufficient assumptions which are stable by conditioning. . < by --V Introduction For real random variables X, Y with df's F, G, define the variability ordering I" /, F <v G if JfdV_< JfdG (I) for all nonincreasing (nondecreasing) convex, integrable functions f in case EX > EY (EX < EY). In particular, forEX = EY (1) holds for all convex, integrable f. If EX < EY, then F <v G is equivalent to E(X- a)+ < E(Y- a)+ for all a ~/R 1. If EX > EY, then F <_~ G is equivalent to E(a - X)+ < E(a - Y)+ for all a ~/R 1. So <v is a combination of a "variability" ordering and of the stochastic ordering <st. In particular, if F <st G or if G <st F, then F <v G. A pure variability ordering is obtained by defining X<vt Y if X-EX<vY-EY. (2) Without preference of the centering at expectations, one could also choose different centering points such as the median. For a general discussion on variability type orderings, we refer to Stoyan [19] and Oja [14]. The converse ordering by nondecreasing concave functions is well established in decision theory in connection with "risk aversion". A sufficient condition for F <~ G is given by S(F- G) < 1, (3) with S the number of sign changes with sign sequence (-, +) in the case of equality. If S(F-G)= 0, then F,G are in a stochastic order relation, while the case S(F- G) = 1 is the well-known cut criterion of Karlin and Novikov. If F r G and 9 J.C. Baltzer AG, Scientific Publishing Company