STOCHASTIC

A UNIFIED APPROACH TO STOCHASTIC DOMINANCE
where
Kt = L - ~ , » = 2, 3, ..., «2" + 1.
In either case, define h„(x) = max;{/!ni(x)}. Then h„ / h and each h„(x) is a
positive linear combination of functions of the form (w — x) + . Define
/«(*) = J? A.O0 4V.
Then /„ is a positive linear combination of functions of the form
Jo i w ~y) + dy = ag(x) + b for some g e G and some constants a and b with
a ;> 0. Since/, / /on [0, oo) and/„ \/on (- oo, 0), it follows that v(/) ^ n(f)
if v(g) ^ fi(g) Vg e G, by applying the monotone and dominated convergence
theorems. The converse is trivial since G ^ U.
III. Probabilistic Content of Stochastic Dominance
In this section an alternative formulation of second-degree stochastic
dominance is presented. This formulation emphasizes the underlying probabilistic
content of the dominance idea. Results are proved only for the case
of bounded random variables.
Suppose that Q a W is a compact convex set in r-dimensional Euclidean
space (e.g., the unit cube). Suppose that X and Y are random variables such
that P[Xea] =P|Tefi] = 1. If v = (v\v 2 , ...,v r ) and v' = (v'\v' 2 , ...,v")
are vectors in W, the notation v ^ v' will be used to denote the r-dimensional
ordering v' g v" (1 ^ i ^ r). A real-valued function / on Q is nondecreasing
if/(u) ^f(v') whenever v ^v'. Let C be the set of all real-valued continuous
functions on Q, and let £P be the set of concave nondecreasing functions on
H. The basic result on stochastic dominance is contained in the following.
Theorem 3.1 The following conditions are equivalent:
(a) Eu(X)^Eu(Y)(ue^).
(b) There exists a random variable Z such that P[Yg, £] = P[_X+Z^ £]
(£, e Q), and E(Z\X) ^ 0 almost surely (i.e., except on a set of probability 0).
The theorem states that X dominates Y if an only if Y is "noisier" than X; for
Y has the same distribution function as X+Z, and Z is random even if a
specific realization of X is given. It should be emphasized that the random
variables Y and X+Z need not be equal even though they have the same
distribution function.
1. STOCHASTIC DOMINANCE 107