Dispersion Comparisons at a Fixed Mean

Dispersion at a Common Mean

Suppose two random variables have the same mean. It is tempting to call the one with larger variance “more dispersed” and stop there. But variance is only one summary: it can conceal where probability lies, and it does not compare every way a distribution can respond to a convex loss. Convex order makes a more comprehensive comparison of dispersion at a fixed mean, while stop-loss expectations reveal exactly what that comparison says about tails.

A second distinction matters. A distribution can be more dispersed in convex order without having a larger probability of exceeding every distance from the mean. Convex order compares integrated tail probabilities, not pointwise tail probabilities. Keeping these claims separate prevents “more dispersed” from being mistaken for “uniformly heavier tails.”

The direction of this notation is worth reading carefully: \(Y\) is at least as dispersed as \(X\) in convex order. For example, every convex penalty, such as a squared-error penalty or a threshold loss, has an expectation no smaller under \(Y\). A convex-order comparison does not say that the distribution functions are ordered at every point.

Stop-Loss Expectations: Integrated Tail Comparisons

For a real threshold \(k\), the stop-loss function of \(X\) is \(E[(X-k)_+]\), where \(u_+=\max(u,0)\). It measures the expected amount by which \(X\) exceeds \(k\). The stop-loss characterization of convex order states that, for integrable \(X,Y\) with equal means, \(X\leq_{\mathrm{cx}}Y\) if and only if \(E[(X-k)_+]\leq E[(Y-k)_+]\) for every real \(k\). Thus checking all thresholds gives a complete comparison, rather than just one variance summary.

To verify the identity, fix a real number \(x\). The length of the set of \(t\geq k\) for which \(x>t\) is exactly \((x-k)_+\). Therefore \((X-k)_+=\int_k^\infty \mathbf{1}_{\{X>t\}}\,dt\) pointwise. The integrand is nonnegative and jointly measurable in the outcome and \(t\), so Tonelli’s theorem permits the exchange of expectation and integration:

Integrability ensures the stop-loss expectation is finite, since \((X-k)_+\leq |X|+|k|\). With equal means, upper and lower stop-loss comparisons contain the same information: for every real \(x,k\), \((x-k)_+-(k-x)_+=x-k\). Taking expectations shows that the difference between the upper and lower stop-loss expectations is \(E[X]-k\), which is the same for \(X\) and \(Y\) when their means agree.

What Variance Does and Does Not Establish

Variance is a convex-order consequence when the second moments are finite, but it is not a substitute for the full comparison. The implication follows by choosing the particular convex function \(\phi(x)=x^2\).

Proof. Convex order applied to \(x^2\) gives \(E[X^2]\leq E[Y^2]\). Because the means are equal, say \(E[X]=E[Y]=m\), and both second moments are finite, \[ \operatorname{Var}(X)=E[X^2]-m^2 \leq E[Y^2]-m^2 =\operatorname{Var}(Y). \] This proves the claim. \(\square\)

The converse does not hold: knowing that \(Y\) has larger variance only compares the expectations of one convex function. Convex order requires the inequality for every convex function—or, equivalently, for the stop-loss function at every threshold.

Convex Order Is Not Pointwise Tail Dominance

One precise meaning of “uniformly heavier two-sided tails” at a common mean \(m\) is that \(P(|Y-m|>t)\geq P(|X-m|>t)\) for every \(t\geq0\). Convex order does not imply this property. Nor is this pointwise condition the stop-loss criterion: the latter integrates one-sided tail probabilities over all thresholds.

The mechanism behind the comparison is also instructive. The values \(-1\) and \(1\) can be viewed as conditional means of more variable outcomes: replace \(-1\) by an equal chance of \(-2\) or \(0\), and replace \(1\) by an equal chance of \(0\) or \(2\). Each replacement preserves its conditional mean while spreading outcomes around it. Convex losses increase under this mean-preserving spread, even though some pointwise tail probabilities decrease.

Using Stop-Loss Comparisons in Practice

A practical comparison at a common mean should begin by fixing the direction of the claim. To show \(X\leq_{\mathrm{cx}}Y\), compare \(E[(X-k)_+]\) with \(E[(Y-k)_+]\) over all real thresholds. A single threshold with the inequality reversed disproves the claim. A variance calculation can quickly disprove a proposed order too: if the proposed convex-order direction contradicts the variance inequality, it cannot hold. But a favorable variance comparison alone is not proof.

Convex order is useful when the question concerns expected convex costs: for example, penalties that grow faster as an outcome moves farther above or below a target. Stop-loss comparisons give a threshold-based way to assess those costs, and the integral identity explains their connection to tail behavior. The conclusion should remain calibrated: convex order establishes greater spread in the convex-function sense, not pointwise dominance of every tail probability.

Common Pitfall

• Wrong reasoning: “The mean is fixed and the variance is larger, so the distribution must be more dispersed in convex order.”
• Correction: A variance comparison checks only the convex function \(x^2\). Convex order requires all convex expectations to be ordered, or equivalently all stop-loss expectations. Test thresholds that could expose a crossing.
• Wrong reasoning: “Convex order means a larger probability of being far from the mean at every distance.”
• Correction: Convex order compares integrated one-sided tails. Pointwise two-sided tail probabilities can cross, as in the preceding example.