An inequality for convex functions involving G-majorization

JOURNAL OF MATHEMATICAL An Inequality AND 69, 603-606 (1979) APPLICATIONS for Convex Functions Involving RAM~N Department ANALYSIS of Statistics, V. G-Majorization” LEON+ AND FRANK PROSCHAN* The Florida State University, Tallahassee, Florida 32306 Submitted by Ky Fan In this paper we derive a simple inequality involving expectations of convex functions and the notion of G-majorization. The result extends a similar inequality of Marshall and Proschan (1965), /. Math. Anal. Applic. Useful applications of the more general inequality are presented. In this note we derive a simple inequality involving expectations of convex functions and the notion of G-majorization. The result extends a similar inequality of Marshall and Proschan (1965) involving majorization. A number of useful applications of the inequality are then presented. Let G be a group of matrices (linear transformations) acting on R’“. A vector a = (n, ,..., a,) is said to G-majorize a vector b = (b, ,..., b,), written a aG b, if b is in the convex hull of the G-orbit of a. If G = P, , the group of permutation matrices, G-majorization coincides with majorization (see Eaton and Perlman, 1976). A random vector X = (Xi ,..., X,) is said to be G-inaariunt if X is stochastically equal to gX for all g E G. When G = P, , we say that X1 ,..., X, are exchangeable random variables. For vectors a and b, let a . b =de* (a& ,..., a,$,). THEOREM I. Let G be a Jinite group such that for all g E G there exist h and k E G for which h(ga . b) = a . kb for all vectors a and b. Let X be a G-invariant random vector, + a continuous, convex, G-invariant function and a aG b. Then &(a . X) > &(b . X). (1) Moreover, if4 is strictly convex, equality holds only when a = gb for some g E G, or when the Xi are all zero with probability one. * The U.S. Government’s right to retain a nonexclusive royalty-free license in and to the copyright covering this paper, for governmental purposes, is acknowledged. + Research supported by National Institute of Environmental Health Sciences under Grant 1 T32 ES07011 at Florida State University. * Research supported by the Air Force dffice of Scientific Research, AFSC, USAF, under Grant AFOSR-74-2581C. 603 0022-247X/79/060603-04$02.00/0 All Copyright 0 1979 by Academic Press, Inc. rights of reproduction in any form reserved. 604 LE6N AND PROSCHAN Proof. Let G = {gi},“4i . Then we may write b = Cy=i aig,a, where each 01~> 0 and Cj”=, + = 1. It follows that &+(b . X) = c?+([C~, ajg,a] . X) 4NZ~1 4w . Xl> < Cz, c4Ww . W. For eachj, let h, and ki be the elements of G for which h,(g,a X) = a . k,X. Then &$(gia . X) = &(hj(gja . X)) [by the G-invariance of $1 = &‘4(a . kjX) = &+(a . X) [by the G-invariance of X]. Thus d’+(b . X) < Cz1 oli&+(a . X) = &4(a . X). In case 4 is strictly convex, it is clear from the above proof that equality holds only if for some g E G, b X = ga X with probability one. 1 Remark 1. If G = P, , then for all g E G and vectors a and b, g-‘(ga b) = a g-lb. Therefore in this special case the hypothesis of Theorem 1 is satisfied. It follows that the main result of Marshall and Proschan (1965) involving majorization is a special case of Theorem 1. Remark 2. Other groups of interest for which the hypothesis of the theorem is satisfied are: (a) The group G, of sign changes and (b) the group G, of permutations and sign changes, as is readily verified. Remark 3. Note that in G, , gpl(ga . b) f a . g-lb for all g E G, . For example if g = [$ -k] then [-y -t]([f -:]a . b) = a . (@ :]b). So in G, the milder requirement is needed that for all g E G there exist h and k E G for which h(ga . b) = a kb. Also note that this condition is not satisfied for some groups. For example, if G = {[-y -:I, I} and g = [-y -:I, then clearly there do not exist h and k for which h(ga . b) = a . kb. Remark 4. Let G = G, and let vectors a and b have all components nonnegative. Then a GG b if and only if a is weakly majorized by b. (See Marshall, Walkup and Wets (1967) for the definition of weak majorization.) Similarly, let G = G, , and let vectors a and b have all components nonnegative. Then a Gc b if and only if ai < bi for i = 1, 2,..., n. It follows that Theorem 1 yields results concerning weak majorization and the usual partial ordering of the plane. (See also Remark 8.) Remark 5. For comments on a converse to Theorem 1, see Remark 3 of Marshall and Proschan (1965). Al so see Remark 4 of that paper for a counterexample showing that the conclusion of Theorem 1 need not necessarily hold when we weaken the hypothesis to require C$to be only continuous and isotone with respect to the G-majorization ordering, i.e., G-monotone. (A G-invariant convex function is necessarily G-monotone). However, by using a path lemma of Eaton and Perlman (1976), I‘t is p ossible to show that if G is a rejlectiongroup, then Theorem 1 holds when 4 is merely continuous and convex along all the line segments joining a with ga for all g E G. (See Eaton and Perlman (1976) for the definition of a reflection group.) This is consistent with Remark 4 of Marshall and Proschan (1965). Th us if G = Ga we need only require that 4, considered as a function of a specified pair of coordinates with all other coordi- AN INEQUALITY 605 FOR CONVEX FUNCTIONS nates held fixed, be convex. Note that this condition on $ is the same as that in Remark 4 of Marshall and Proschan (1965). Similarly, if G = G, , we need only require that $, considered as a function of a specified coordinate with all other coordinates held fixed, be convex. COROLLARY 1. Let G = G, . Let X(U,),..., X(a,) be independent random variables, where X(q) is normally distributed with mean zero and standard deviation cri . Let + be continuous, convex, and invariant under permutations and sign changes,and let (aI ,..., a,) aG (CT:,..., uh). Then ~#q~l),..., X(4 > 4w(4,..., X(4,. (2) Proof. Let YI ,..., Y, be independently distributed standard normal random variables. Then @(ulYl ,..., u,Y,) > cF$(u;Y, ,..., uLY,J by Theorem 1. Since u,Yi and X(uJ have the same distribution, the result follows. 1 Remark 6. Similar results are true when G is P, or G, . Remark 7. Note that the only property of X(uJ used in the proof of Corollary 1 is that X(U,) and uiY have the same distribution where Y is a random variable distributed symmetrically about zero. Thus, for example, Corollary 1 is still true when X(ui) is uniformly distributed on the interval (-ui , ui). For simplicity, Corollary 1 is stated for the special case X(ui) is normal. Remark 8. Note that since a dpn b or a <cl b implies a bGz b, (2) holds when a Gpn b or a <cl b. (A similar remark applies whenever the G, ordering holds.) Some well known results may be obtained immediately from Theorem 1; in addition, an extension will be presented in Corollary 4: COROLLARY 2 (Karamata). If 4 is continuous and convex, a bP* b, then C #(ail G C (3) 9W~ Proof. Apply the theorem to the case #(x1 ,..., x,) = C 4(x$) and G = P, with P((X, ,..., X,) = ei> = l/n, i = I ,..., n, where ei is the vector with one in the ith place and zeros elsewhere. 1 This inequality is given in Hardy, Littlewood, and Polya (1952), p. 89. COROLLARY a 3. If yi > 0, i = l,,.., n, and a <Pe b, then (Muirhead) I! yFr,“z ... y; <I! y;yz . . . y>, where x! denotessummation over the n! permutations of the yi . (4) 606 LE6N AND PROSCHAN More generally, (Marshall and Proschan) I! $(aIxl ,..., a,x,) < I! 4(6,x, ,..., b,x,), (5) where 4 is continuous, convex, and symmetric, and C! denotes summation over the n! permutations of the xi . Proof. If G = P,, , (5) is an immediate consequence of (l), where pm ,**.,XT%) = (Xi,,...,Xi”>> =f whenever (il ,..., i,) is a permutation of (1,2 ,..., n). With $(x1 ,.. ., x,) = exp(C xi) and xi = log yi , (5) yields (4). i COROLLARY 4. Let 4 be continuous, convex, and invariant and sign changes, and let a aGa b. Then under permutations C*+(wI,...,a,4 2 c*%x1,..-,b,x,), where C* denotes summation over all sign changes and permutations (6) of the xi . (6) is an immediate consequence of (l), where P{(X, ,..., X,) = xi,) = l/2%!, ,where (i1 ,..., i,) is a permutation of y21F xi1 ,..., (-lp* n) and (TV = 0 or 1 for i = l,..., n. >...> Proof. Remark 9. For other possible applications yielding inequalities, see Marshall and Proschan (1965). REFERENCES J. C. CONLON, R. LEON, F. PROSCHAN, AND J. SETHURAMAN, “G-Ordered Functions, with Applications in Statistics. I. Theory,” AFOSR Report 74, Department of Statistics, Florida State University, 1977. J. C. CONLON, R. LEON, F. PROSCHAN, AND J. SETHUFXAMAN, “G-Ordered Functions, AFOSR Report 75, Department of with Applications in Statistics. II, Applications,” Statistics, Florida State University, 1977. M. L. EATON AND M. D. PERLMAN, “Reflection Groups, Generalized Schur Functions, University of Chicago Statistics Report No. 28, and the Geometry of Majorization,” 1976. G. H. H-Y, J. E. LITTLEWOOD, AND G. P~LYA, “Inequalities,” Cambridge Univ. Press, London/New York, 1952. K. JOGDEO, Association and probability inequalities, Ann. Statist. 5 (1977), 495-504. A. W. MARSHALL AND F. PROSCHAN,An inequality for convex functions involving majorization, J. Math. Anal. Appl. 12 (1965), 87-90. A. W. MARSHALL, D. W. WALKUP, AND R. 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