These authors contributed equally to this work.

This paper describes a complementary tool for fitting probabilistic distributions in data analysis. First, we examine the well known bivariate index of skewness and the aggregate skewness function, and then introduce orderings of the skewness of probability distributions. Using an example, we highlight the advantages of this approach and then present results for these orderings in common uniparametric families of continuous distributions, showing that the orderings are well suited to the intuitive conception of skewness and, moreover, that the skewness can be controlled via the parameter values.

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 [

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) [

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) [

Many authors have proposed and obtained different descriptive elements to measure skewness (see, for instance, [

Ref [

García et al. (2015) [

The distances,

The relationship

If

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

As it is easily proven that

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 [

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) [

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

Let

With these definitions, it immediately follows that:

The proof follows immediately from the definitions given in (

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.

Let

Part 1. We can write

Part 2. For

Then, clearly we have that

The CDF of a uniparametric log–logistic distributed random variable

Notice that

When

As

Let

If we consider

For

Firstly, for

Secondly, for

Finally, when

Hence, the proof is completed. ☐

For

Consider the uniparametric Weibull distributions family given by the CDF

The mode is known to be at 0, for

On the one hand, when

For

The asymmetric Laplace distribution has been introduced in the literature by different ways ([

The aggregate skewness function of an

At

As a conclusion, we can enunciate the following Proposition, whose proof is straightforward and hence omitted.

The methods for Project Management and Review Technique (PERT) are well known and widely applied when the needed activities for a given project must be ordered according to precedence in time. Some of these methods require modelling the time length of each activity as a random variable, following an expert’s opinion. The beta and the asymmetric triangular distributions are commonly used by engineers to describe these time lengths. In any case, the indications of the experts can be related to a maximum and a minimum values and a mode, often completed with further considerations about the shape and skewness of the PDF of the time random variable. Then, a deep study of the skewness of both families of probability distributions would be welcome to improve the model fit.

On the one hand, the asymmetric standard triangular distribution (ASTD) , free of center and scale parameters, depends on only one parameter

There is a large body of literature that shows the use of the ASTD in PERT methods (see [

For

As the mode is found to be at

In the case

Some algebra allows to prove that, being

If

If

Therefore, the skewness of the ASTD distributions is completely controlled by the parameter

On the other hand, the pdf of a beta distribution is given by

We are interested on the cases

Hence, we only consider cases where

Notice that

Firstly, observe that

Secondly, if

Hence, for

Now we focus on the family of Beta distributions with given mode,

First case, the constrains are

Second case, the constrain are

With these results, we can conclude that

In this paper two main objectives are achieved: on the one hand, the given examples show that the skewness function orders the mesh in good accordance with the intuitive conception of skewness. Moreover, these examples show that the skewness of a distribution obtained from certain parametric families can be controlled by reference to their parameters.

As we show, the function

In practice, much can be learned from this model, but there remains the risk that it may be wrongly specified in real applications. Thus, in practice we must be willing to assume that the underlying distribution has a unique mode and belongs to a uniparametric family of distributions.

In many practical situations, the maximum skewness index coincides with the well known

All authors have contributed equally to this paper.

This research received no external funding.

This research was partially funded by MINECO (Spain) grant number EC02017–85577–P. The authors are grateful for helpful suggestions made by two reviewers.

The authors declare no conflict of interest.

Skewness functions

Beta distributions with common given mode