# A Note on Ordering Probability Distributions by Skewness

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## Abstract

**:**

## 1. Introduction

- $\left({S}^{-}\left(F\right),{S}^{+}\left(F\right)\right)=\mathbf{0},$ for any symmetric distribution F.
- $\left({S}^{-}\left(aF+b\right),{S}^{+}\left(aF+b\right)\right)=\left({S}^{-}\left(F\right),{S}^{+}\left(F\right)\right)$, for all $a>0$, $-\infty <b<\infty $.
- $\left({S}^{-}\left(-F\right),{S}^{+}\left(-F\right)\right)=\left(-{S}^{+}\left(F\right),-{S}^{-}\left(-F\right)\right)$.
- If $F{<}_{c}G$, then $\left({S}^{-}\left(F\right),{S}^{+}\left(F\right)\right)\le \left({S}^{-}\left(G\right),{S}^{+}\left(G\right)\right),$ understood as vector dominance.

**Example**

**1.**

#### Outline

## 2. Families of Uniparametric Distributions Ordered by Skewness

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

#### 2.1. Uniparametric Gamma Distributions

**Proposition**

**2.**

- 1.
- If $-1<{\alpha}_{1}<{\alpha}_{2}<0$, then $G\left({\alpha}_{2}\right){\ge}_{\nu}G\left({\alpha}_{1}\right).$
- 2.
- If $0<{\alpha}_{1}<{\alpha}_{2}$, then $G\left({\alpha}_{1}\right){\ge}_{+}G\left({\alpha}_{2}\right).$

**Proof.**

#### 2.2. Log–Logistic Distributions

**Proposition**

**3.**

**Proof.**

#### 2.3. Lognormal Variance Distributions

**Proposition**

**4.**

**Proof.**

#### 2.4. Uniparametric Weibull Distributions

**Proposition**

**5.**

**Proof.**

#### 2.5. Asymmetric Laplace Distributions

- The aggregate skewness function of an $AL\left(\kappa \right)$ distribution can be written as$${\nu}_{AL}\left(z;\kappa \right)=\frac{1}{1+{\kappa}^{2}}\left[\mathrm{exp}\left(-\sqrt{2}\kappa z\right)-{\kappa}^{2}\mathrm{exp}\left(-\frac{\sqrt{2}}{\kappa}z\right)\right].$$
- ${\nu}_{AL}\left(z;\kappa \right)$ is an increasing negative function of z when $\kappa >1$, and it is a decreasing positive function of z when $0<\kappa <1$. ${\nu}_{AL,1}\left(z;1\right)=0$, for all $z\ge 0$. That is, any $AL$ distribution is skewed only to the right or to the left, depending on $\kappa $. In any case, the function verifies ${lim}_{z\to \infty}{\nu}_{AL}\left(z;\kappa \right)=0$ but, when $\kappa \ne 1$, the function never reaches that limit value. To prove these results, it is sufficient to note that$$\frac{d{\nu}_{AL}\left(z;\kappa \right)}{dz}=\frac{\sqrt{2}\kappa}{{\kappa}^{2}+1}\left[\mathrm{exp}\left(-\frac{\sqrt{2}}{\kappa}z\right)-\mathrm{exp}\left(-\sqrt{2}\kappa z\right)\right].$$
- At $z=0$, the skewness function takes the following value:$${\nu}_{AL}\left(0;\kappa \right)=\frac{1-{\kappa}^{2}}{1+{\kappa}^{2}}.$$
- ${\nu}_{AL}\left(z;\kappa \right)$ is a strictly decreasing function on $\kappa $. This is easily shown by means of$$\frac{d{\nu}_{AL}\left(z;\kappa \right)}{d\kappa}=-\frac{\sqrt{2}z{\kappa}^{2}+2\kappa +\sqrt{2}z}{{\left({\kappa}^{2}+1\right)}^{2}}\left[\mathrm{exp}\left(-\sqrt{2}\frac{z}{\kappa}\right)+\mathrm{exp}\left(-\sqrt{2}\kappa z\right)\right]<0,$$

**Proposition**

**6.**

- 1.
- ${F}_{AL}\left({\kappa}_{1}\right){\ge}_{\nu}{F}_{AL}\left({\kappa}_{2}\right).$
- 2.
- If $0<{\kappa}_{1}<1$, then ${F}_{AL}\left({\kappa}_{1}\right)$ is skewed only to the right.
- 3.
- If ${\kappa}_{1}>1$, then ${F}_{AL}\left({\kappa}_{1}\right)$ is skewed only to the left.

## 3. The Beta and the AST Distributions

- $ASTD\left({\theta}_{1}\right){\ge}_{\nu}ASTD\left({\theta}_{2}\right).$
- If $0<{\theta}_{i}<0.5$, then ${S}_{ASTD}^{+}\left({\theta}_{i}\right)={\nu}_{ASTD}\left(0;{\theta}_{i}\right)=1-2{\theta}_{i}$, and ${S}_{ASTD}^{-}\left({\theta}_{i}\right)=0$.
- If $0.5<{\theta}_{i}<1$, then ${S}_{ASTD}^{+}\left({\theta}_{i}\right)=0$ and ${S}_{ASTD}^{-}\left({\theta}_{i}\right)={\nu}_{ASTD}\left(0;{\theta}_{i}\right)=1-2{\theta}_{i}$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Beta distributions with common given mode $M=0.2$ (left panel) for $\alpha =2,4$ and 9 and their skewness functions ${\nu}_{B}$ (right panel).

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**MDPI and ACS Style**

García, V.J.; Martel-Escobar, M.; Vázquez-Polo, F.J. A Note on Ordering Probability Distributions by Skewness. *Symmetry* **2018**, *10*, 286.
https://doi.org/10.3390/sym10070286

**AMA Style**

García VJ, Martel-Escobar M, Vázquez-Polo FJ. A Note on Ordering Probability Distributions by Skewness. *Symmetry*. 2018; 10(7):286.
https://doi.org/10.3390/sym10070286

**Chicago/Turabian Style**

García, V. J., M. Martel-Escobar, and F. J. Vázquez-Polo. 2018. "A Note on Ordering Probability Distributions by Skewness" *Symmetry* 10, no. 7: 286.
https://doi.org/10.3390/sym10070286