**1. Introduction**

Detailed knowledge of the characteristics of probability models is desirable (if not essential) if data are to be modeled properly. In studying these properties, many authors have considered orderings within probability distribution families, according to diverse measuring criteria. The usual approach taken by researchers in this field is to evaluate or measure one or more theoretical characteristics of a given distribution and to study the effect produced by the value of its parameters on this measurement. In actuarial science, stochastic orders are widely used in order to make risk comparisons [1].

Some parametric distributions can be ordered according to the evaluation made of a given property, merely by comparing some of its parameters. Although most related orders are actually preorders, each one presents interesting applications. Many studies have been conducted in this area, and the following are particularly significant: Lehmann (1955) [2], which is of seminal importance; Arnold (1987) [3], who compared random variables according to stochastic ordering in a particular Lorenz order; Shaked and Shanthikumar (2006) [1], on stochastic orders; Nanda and Shaked (2001) [4], on reversed hazard rate orders; Ramos-Romero and Sordo-Díaz (2001) [5], on the likelihood ratio order; and Gupta and Aziz (2010) [6], on convex orders.

In this paper, we study the relationship between the skewness of some parametric distributions and the value of one of their parameters. The first question to be addressed is that of measuring the skewness. In this respect, Oja (1981) [7] introduced a set of axioms to be verified by any measurement of skewness considered. These axioms were established for indexes of skewness with one main constraint: that the skewness of a distribution should be evaluated by a single real number. This point is discussed below.

Many authors have proposed and obtained different descriptive elements to measure skewness (see, for instance, [8–13]). Ref [10] suggested a measurement of skewness corresponding to the (unique) mode, *M*, given by the following index:

$$\gamma\_M\left(F\right) = 1 - 2F\left(M\right). \tag{1}$$

Ref [10] applied this index to ordering the gamma, log-logistic, lognormal and Weibull families of distributions by their skewness, taking into account the feasible values of their respective parameters. Index (1), which is proven to satisfy those axioms derived from Oja (1981) [7], is also recommended in [14] as a (very) good index of skewness. However, notice that (1) only compares the probability weight on the left side of a central point (the mode) with the value 1/2, but it does not account for how the weights are distributed to each side of the centre.

García et al. (2015) [15] introduced some further elements to be incorporated into the list of skewness measurements of a probability distribution. According to these authors, given a unimodal probability distribution *F* (*x*), its skewness is considered to be a local function of a given distance, *z*, from the mode, *M*. For such a distance, and given the interval [*M* − *z*, *M* + *z*], the aggregate skewness function, *νF* (*z*), compares the probability weight of *F* at either side of the interval:

$$\nu\_{\mathbb{F}}\left(z\right) = \Pr\left(X > M + z\right) - \Pr\left(X < M - z\right),\tag{2}$$

where *z* ≥ 0. Thus, the (maximum) right skewness of the distribution *F* and its (minimum) left skewness are respectively given by

$$S^{+}\left(F\right) = \max\_{z \ge 0} \nu\_F\left(z\right), \quad S^{-}\left(F\right) = \min\_{z \ge 0} \nu\_F\left(z\right). \tag{3}$$

The distances, *zp* and *zn*, where these extreme values are achieved, are termed the critical distances to the mode. As the skewness function is bounded inside the interval [−1, 1] and *νF* (∞) = 0, the bivariate index *<sup>S</sup>*− (*F*), *S*<sup>+</sup> (*F*) belongs to [−1, 0] × [0, 1]. A given distribution function *F* such that *νF* (*z*) ≥ 0 for all *z* ≥ 0 is said to be only skewed to the right; and if *νF* (*z*) ≤ 0 for all *z* ≥ 0, it is said to be only skewed to the left.

The relationship *F* <*c G* (*F <sup>c</sup>*−precedes *G*) means that *G*−<sup>1</sup> [*F* (*x*)] is a convex function. For a continuous distribution *F*, the bivariate measurement of skewness *<sup>S</sup>*− (*F*), *S*<sup>+</sup> (*F*) verifies the following properties, where *aF* + *b* and −*F* mean the distributions of the corresponding transformation of a random variable that is *F*−distributed:


These properties can be considered as a vectorial interpretation of the axioms given by Oja (1981) [7].

As it is easily proven that *νF* (0) = *γM* (*F*), we can establish that (2) and (3) give considerably clearer and more complete information than (1) about the skewness of any distribution function.

Most families of continuous distributions are only skewed to the right (or only to the left), while doubles-sign skewness is abundant within the discrete families, as shown in [15]. Nevertheless, the joint use of the function (2) and the bivariate index (3) makes it possible to improve the ordering of the skewness-based distribution discussed in [10], as can be seen in the following example.

**Example 1.** *Assume the following random variable X* ∈ [−2, ∞) *with PDF given by:*

$$f'(x) = \begin{cases} \frac{1}{4}x + \frac{1}{2}, & -2 \le x < 0; \\\\ \frac{1}{2}\exp\left(-x\right), & x \ge 0. \end{cases}$$

*Assume also the PDF, g* (*y*) *of Y* = −*X. Then, γM* (*F*) = 0 = *γM* (*G*). *That is, according to coefficient γM* (·)*, both distributions have the same null skewness, although they do not even have a symmetric support set.*

*However, using expression (2), we find that*

$$\nu\_F(z) = \begin{cases} \frac{1}{2} \exp\left(-z\right) - \frac{1}{8} \left(z - 2\right)^2, & 0 \le z < 2; \\\\ \frac{1}{2} \exp\left(-z\right), & z \ge 2; \end{cases}$$

*and νG* (*z*) = −*νF* (*z*), *for all z* ≥ 0*. These functions are plotted in Figure 1, where it can be seen that νF* (*z*) ≥ *νG* (*z*) *for all z* ≥ 0*, S*<sup>+</sup> (*F*) = −*S*<sup>−</sup> (*G*) = 0 *and S*− (*F*) = 0 = *S*<sup>+</sup> (*G*)*. Clearly, the information about skewness obtained from the aggregate skewness function ν* (*z*) *and the indices S*<sup>+</sup> (·) *and S*− (·) *is considerably more comprehensive than that obtained from γM* (·)*.*

**Figure 1.** Skewness functions *<sup>ν</sup>F*(*z*) and *<sup>ν</sup>G*(*z*) in Example 1.
