*Outline*

In applied statistical analysis, it is useful to have a large catalogue of plausible distributions with which to fit the data. According to García et al. (2015) [15], common measures of skewness can be complemented with a bivariate index of positive–negative skewness, and the authors show that the mode is the relevant central value to study both right and left skewness. In this paper, we extend the tool-box approach to fit data from probability distributions, introducing two orderings that are deduced from the skewness measures given in [15]. The first of those orderings is based on the positive part of the bivariate index of skewness, which in many instances coincides with the well known *γM* (*F*). Nevertheless, the differences can be highly significant, as in the previous example. The second, more

noteworthy, order is based on the skewness function *νF* (*z*) and meets the first of the conditions, but not the reciprocal.

There are two reasons for ordering a family of distributions according to a given measurement of skewness. Firstly, as a property of the distribution, this ordering allows us to control its skewness by the appropriate selection of the parameter. When this is done (and the parameter is readily determined), the theoretical results have immediate applications in the data-fitting process. Secondly, when a given family of distributions is conceived as being more or less skewed according to the value of a parameter, and a measurement of skewness ratifies the ordering, it may be concluded that the functioning of this measurement provides a reasonably good fit with an intuitive conception of skewness.

The rest of this paper is organized as follows. In Section 2, we study the aggregate skewness function and the resultant skewness-based ordering of the gamma, log-logistic, lognormal, Weibull and asymmetric Laplace families of continuous probability distributions. In Section 3, we study the ordering of two of the most well-known distributions commonly used in PERT methods: the beta and the asymmetric triangular distributions. Finally, conclusions are presented in Section 4.

### **2. Families of Uniparametric Distributions Ordered by Skewness**

Let *F* and *G* be unimodal distribution functions, with no centre or scale parameters, and modes *MF* and *MG*, respectively. We compare their respective skewness by two different criteria.
