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

Abstract

Let S₁ and S₂ be two signals of a random variable X, where G₁(s₁ ∣ x) and G₂(s₂ ∣ x) are their conditional distributions given X = x. If, for all s₁ and s₂, G₁(s₁ ∣ x) - G₂(s₂ ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S₁ is greater than the conditional expectation of X given S₂ 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 S₁ and S₂ have the same marginal distribution and, when S₁ and S₂ are sums of X and independent noises, it is equivalent to the dispersive order of the noises.