*2.5. Asymmetric Laplace Distributions*

The asymmetric Laplace distribution has been introduced in the literature by different ways ([16,17]). In this paper we will use Kozubowski and Podgórski (2002) [18] (later refined in [19]) to refer it. This distribution is obtained by using the scheme introduced by Fernández and Steel (1998) [20] to produce skewness on a symmetric distribution. In this way, the pdf of a skewed or asymmetric Laplace distribution can be written in the form

$$f\left(\mathbf{x};\mu,\sigma,\kappa\right) = \begin{cases} \begin{array}{c} \frac{\sqrt{2}}{\sigma} \frac{\kappa}{1+\kappa^2} \exp\left[-\frac{\sqrt{2}}{\kappa\sigma}\left(\mu-\mathbf{x}\right)\right], \quad \mathbf{x}<\mu, \\\ \frac{\sqrt{2}}{\sigma} \frac{\kappa}{1+\kappa^2} \exp\left[-\frac{\kappa\sqrt{2}}{\sigma}\left(\mathbf{x}-\mu\right)\right], \quad \mathbf{x}\ge\mu, \end{array} \end{cases}$$

where *σ*, *κ* > 0, and −∞ < *μ* < ∞. Then, we assign values (0, 1) to the centre and scale parameters (*μ* and *σ*, respectively) in order to study the aggregate skewness function, and the extreme right and left skewness indices then depend only on the skewness parameter *κ* > 0. Thus, it is easily proven that:

1. The aggregate skewness function of an *AL* (*κ*) distribution can be written as

$$\nu\_{AL}\left(z;\kappa\right) = \frac{1}{1+\kappa^2} \left[ \exp\left(-\sqrt{2}\kappa z\right) - \kappa^2 \exp\left(-\frac{\sqrt{2}}{\kappa} z\right) \right].$$

2. *νAL* (*z*; *κ*) is an increasing negative function of *z* when *κ* > 1, and it is a decreasing positive function of *z* when 0 < *κ* < 1. *<sup>ν</sup>AL*,<sup>1</sup> (*z*; 1) = 0, for all *z* ≥ 0. That is, any *AL* distribution is skewed only to the right or to the left, depending on *κ*. In any case, the function verifies lim*z*→∞ *νAL* (*z*; *κ*) = 0 but, when *κ* = 1, the function never reaches that limit value. To prove these results, it is sufficient to note that

$$\frac{d\boldsymbol{v}\_{AL}(\boldsymbol{z};\boldsymbol{\kappa})}{d\boldsymbol{z}} = \frac{\sqrt{2}\boldsymbol{\kappa}}{\kappa^2 + 1} \left[ \exp\left(-\frac{\sqrt{2}}{\kappa}\boldsymbol{z}\right) - \exp\left(-\sqrt{2}\boldsymbol{\kappa}\boldsymbol{z}\right) \right].$$

3. At *z* = 0, the skewness function takes the following value:

$$\nu\_{AL}\left(0; \kappa\right) = \frac{1 - \kappa^2}{1 + \kappa^2}.$$

Then, *νAL* (0; *κ*) is the value for *S*<sup>+</sup> (*FAL* (*κ*)) or *S*− (*FAL* (*κ*)), depending on its sign. 4. *νAL* (*z*; *κ*) is a strictly decreasing function on *κ*. This is easily shown by means of

$$\frac{d\nu\_{AL}(z;\kappa)}{d\kappa} = -\frac{\sqrt{2}z\kappa^2 + 2\kappa + \sqrt{2}z}{\left(\kappa^2 + 1\right)^2} \left[ \exp\left(-\sqrt{2}\frac{z}{\kappa}\right) + \exp\left(-\sqrt{2}\kappa z\right) \right] < 0,$$

for all *z* > 0, and all *κ* > 0.

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

**Proposition 6.** *Assume* 0 < *κ*1 < *κ*2 < <sup>∞</sup>*, and let FAL* (*<sup>κ</sup>*1) *and FAL* (*<sup>κ</sup>*2) *be the respective asymmetric Laplace distributions. Then:*


### **3. The Beta and the AST Distributions**

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 0 ≤ *θ* ≤ 1, and has the pdf:

$$f\left(\mathbf{x}|\theta\right) = \begin{cases} \begin{array}{c} 2\mathbf{x}\theta^{-1}, & 0 \le \mathbf{x} \le \theta, \\ 2\left(1-\mathbf{x}\right)\left(1-\theta\right)^{-1}, & \theta \le \mathbf{x} \le 1, \\ 0, & \text{elsewhere.} \end{array} \right)$$

There is a large body of literature that shows the use of the ASTD in PERT methods (see [21] and [19] and cites therein). Note that cases *θ* = 0, 1 are members of the beta family of distributions.

For 0 < *θ* < 1, the *ASTD*(*θ*) CDF can be written as follows:

$$F\left(\mathbf{x}|\theta\right) = \begin{cases} \mathbf{x}^2 \theta^{-1} , & 0 \le \mathbf{x} \le \theta \\\ \left(2\mathbf{x} - \mathbf{x}^2 - \theta\right) \left(1 - \theta\right)^{-1} , & \theta \le \mathbf{x} \le 1 \end{cases}$$

As the mode is found to be at *x* = *θ*, its skewness function is found to be

$$\nu\_{ASTD}\left(z;\theta\right) = \left(1 - 2\theta\right) - \frac{\left(1 - 2\theta\right)}{\theta\left(1 - \theta\right)}z^2.$$

for 0 ≤ *z* ≤ min {*<sup>θ</sup>*, 1 − *θ*} . In the case *θ* = 0.5, the skewness function is null. Then, for 0 < *θ* < 0.5 and *θ* < *z* ≤ 1 − *θ*,

$$\nu\_{\rm ASTD} \left( z; \theta \right) = \frac{\left( z - 1 + \theta \right)^2}{1 - \theta}.$$

In the case 0.5 < *θ* < 1, for 1 − *θ* < *z* ≤ *θ*,

$$\nu\_{ASTD}\left(z;\theta\right) = -\frac{\left(\theta - z\right)^2}{\theta}\prime$$

and it is easily found that

$$\nu\_{\rm ASTD} \left( z; \theta \right) = -\nu\_{\rm ASTD} \left( z; 1 - \theta \right) \, , \tag{13}$$

for 0 ≤ *z* < ∞.

> Some algebra allows to prove that, being 0 < *θ*1 < *θ*2 < 1,


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

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

$$f\_B\left(\mathfrak{x}; \mathfrak{a}, \beta\right) = \frac{\mathfrak{x}^{\mathfrak{a}-1} \left(1 - \mathfrak{x}\right)^{\beta - 1}}{B\left(\mathfrak{a}, \beta\right)}, \qquad 0 \le \mathfrak{x} \le 1, \beta$$

where *α*, *β* > 0, and *B* (·, ·) is the beta function. Given that its CDF *F* (*x*; *α*, *β*) verifies that *F* (*x*; *α*, *β*) = 1 − *F* (*x*; *β*, *α*) and the sign of its skewness depends only on the condition *β* ≥ *α* or *β* ≤ *α*, we can study only the case *β* > *α*.

We are interested on the cases *α*, *β* > 1, where there is an unique mode *M*,

$$M = \frac{\alpha - 1}{\alpha + \beta - 2} \doteq b \left( \alpha, \beta \right).$$

Hence, we only consider cases where 1 < *α* < *β*, where there exists a right skewness; the cases 1 < *β* < *α*, with left skewness, can be immediately deducted by taking the parameters in reverse.

Notice that Pr(*X* > *M* + *z*) > 0 requires 0 ≤ *z* ≤ *b* (*β*, *<sup>α</sup>*), and that Pr(*X* < *M* − *z*) > 0 requires 0 ≤ *z* ≤ *b* (*<sup>α</sup>*, *β*). Then,

$$\nu\_B\left(z;a,\beta\right) = \begin{cases} 1 - I\_{M+z}\left(a,\beta\right) - I\_{M-z}\left(a,\beta\right) > 0, & 0 \le z \le b\left(a,\beta\right) \\ \qquad 1 - I\_{M+z}\left(a,\beta\right) > 0, & b\left(a,\beta\right) < z \le b\left(\beta,a\right) \\ \qquad 0, & z > b\left(\beta,a\right) \end{cases} \tag{14}$$

where,

$$I\_z\left(\alpha,\beta\right) = \int\_0^z \frac{t^{\alpha-1} \left(1-t\right)^{\beta-1} dt}{B\left(\alpha,\beta\right)}$$

is the well known Beta Regularized function.

Firstly, observe that

$$\nu\_B\left(0; \alpha, \beta\right) = 1 - \frac{2}{B\left(\alpha, \beta\right)} \int\_0^M \mathbf{x}^{\alpha - 1} \left(1 - \mathbf{x}\right)^{\beta - 1} d\mathbf{x}\_{\alpha}$$

and

$$\frac{1}{B\left(\alpha,\beta\right)}\int\_0^M \mathbf{x}^{a-1} \left(1-\mathbf{x}\right)^{\beta-1} d\mathbf{x} < \frac{1}{B\left(\alpha,\beta\right)}\int\_0^m \mathbf{x}^{a-1} \left(1-\mathbf{x}\right)^{\beta-1} d\mathbf{x} \approx \frac{1}{2}\sqrt{\alpha}$$

where

$$m = \frac{\alpha - \frac{1}{3}}{\alpha + \beta - \frac{2}{3}}$$

is the approximate median of the distribution.

Secondly, if 0 ≤ *z* ≤ *b* (*<sup>α</sup>*, *β*) < *b* (*β*, *<sup>α</sup>*), then

$$B\left(a,\beta\right) \cdot \frac{d\nu\_{\mathbb{B}}\left(z;a,\beta\right)}{dz} = -\left[b\left(a,\beta\right) + z\right]^{a-1}\left[b\left(\beta,a\right) - z\right]^{\beta-1} - \left[b\left(a,\beta\right) - z\right]^{a-1}\left[b\left(\beta,a\right) + z\right]^{\beta-1},$$

which is negative within the rank of *z*. For *b* (*<sup>α</sup>*, *β*) < *z* ≤ *b* (*β*, *<sup>α</sup>*),

$$B\left(\mathfrak{a},\mathfrak{f}\right) \cdot \frac{d\nu\_B\left(z;\mathfrak{a},\mathfrak{f}\right)}{dz} = -\left[b\left(\mathfrak{a},\mathfrak{f}\right) + z\right]^{a-1} \left[b\left(\mathfrak{f},\mathfrak{a}\right) - z\right]^{\mathfrak{f}-1} < 0.$$

Hence, for 0 ≤ *z* ≤ *b* (*β*, *<sup>α</sup>*), *νB* (*z*; *α*, *β*) is a strictly decreasing continuous function with *νB* (0; *α*, *β*) > 0 and *νB* (*b* (*β*, *α*); *α*, *β*) = 0.

Now we focus on the family of Beta distributions with given mode, *M*. That is, we consider the subfamily of Beta distributions:

$$B\left(\alpha+1, 1+\frac{1-M}{M}\alpha\right),$$

with *α* > 0. Then, with the aid of a proper software (we have used Wolfram Mathematica 10), one can obtain the derivative 

$$\frac{\partial}{\partial \alpha} \nu\_B \left( z; \alpha + 1, 1 + \frac{1 - M}{M} \alpha \right) \dots$$

and maximize this function, in two cases:

First case, the constrains are *α* ≥ 1, 0 < *m* < 1/2, 0 ≤ *z* ≤ *b* (*α* + 1, 1 + (1 − *M*) *<sup>α</sup>*/*M*). The maximum value of the function is 0, and it is achieved when *M* = 0.5, *α* 3.54147, *z* 0.309936.

Second case, the constrain are *α* ≥ 1, 0 < *m* < 1/2, *b* (*α* + 1, 1 + (1 − *M*) *α*/*M*) < *z* ≤ *b* (1 + (1 − *M*) *<sup>α</sup>*/*M*, *α* + <sup>1</sup>). The maximum value of the function is −5.07056 × <sup>10</sup>−6, and it is achieved when *M* = 0.123564, *α* 1.62726, *z* 0.632457.

With these results, we can conclude that *νB* (*z*; *α* + 1, 1 + (1 − *M*) *α*/*M*) decreases with the feasible values of *α*. That way, the subsets of Beta distributions with fixed mode are ordered on skewness (see Figure 2). As the parameter values increase, these Beta distributions become less skewed.

**Figure 2.** Beta distributions with common given mode *M* = 0.2 (left panel) for *α* = 2, 4 and 9 and their skewness functions *νB* (right panel).
