A relation between positive dependence of signal and the variability of conditional expectation given signal

摘要

Let S-1 and S-2 be two signals of a random variable X, where G(1)(s(1) vertical bar x) and G(2)(S-2 vertical bar x) are their conditional distributions given X = x. If, for all s(1) and s(2), G(1)(s(1) vertical bar x) - G(2)(s(2) vertical bar x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S-1 is greater than the conditional expectation of X given S-2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S-1 and S-2 have the same marginal distribution and, when S-1 and S-2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.