**Definition 1.** *If*

$$
\nu\_{\mathbb{F}}\left(z\right) - \nu\_{\mathbb{G}}\left(z\right) \ge 0, \qquad \forall z \ge 0,\tag{4}
$$

*then we say that F has equal or more aggregate skewness to the right at any point than G*. *We denote this by F* ≥*ν G.*

**Definition 2.** *If F and G are both skewed only to the right, we say that F has equal or more maximum aggregate skewness to the right than G when*

$$S^{+}\left(F\right) - S^{+}\left(G\right) \geq 0,\tag{5}$$

*and we denote this by F* ≥+ *G.*

> With these definitions, it immediately follows that:

### **Proposition 1.** *If F* ≥*ν G*, *then F* ≥+ *G*.

*The reverse implication is not true in general.*

**Proof.** The proof follows immediately from the definitions given in (4) and (5).

In the next section, we consider some well known uniparametric families of continuous distributions, with no centre or scale parameters but depending on a skewness parameter, and examine whether they are ordered by aggregate skewness, or by maximum aggregate skewness. The gamma family is a very broad one, which includes many other well known distributions as particular cases. A study of the log-logistic, lognormal, Weibull and asymmetric Laplace families, one by one and in turn, when not included inside the previous one, will produce widely varying results.
