# A Note on Ordering Probability Distributions by Skewness

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## Abstract

**:**

## 1. Introduction

- $\left(\right)open="("\; close=")">{S}^{-}\left(F\right),{S}^{+}\left(F\right)=\mathbf{0},$ for any symmetric distribution F.
- $\left(\right)open="("\; close=")">{S}^{-}\left(\right)open="("\; close=")">aF+b,{S}^{+}\left(\right)open="("\; close=")">aF+b}=\left(\right)open="("\; close=")">{S}^{-}\left(F\right),{S}^{+}\left(F\right)$, for all $a>0$, $-\infty <b<\infty $.
- $\left(\right)open="("\; close=")">{S}^{-}\left(\right)open="("\; close=")">-F,{S}^{+}\left(\right)open="("\; close=")">-F}=\left(\right)open="("\; close=")">-{S}^{+}\left(F\right),-{S}^{-}\left(\right)open="("\; close=")">-F$.
- If $F{<}_{c}G$, then $\left(\right)open="("\; close=")">{S}^{-}\left(F\right),{S}^{+}\left(F\right)\le \left(\right)open="("\; close=")">{S}^{-}\left(G\right),{S}^{+}\left(G\right)$ 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 $G\left(\right)open="("\; close=")">{\alpha}_{2}.$
- 2.
- If $0<{\alpha}_{1}<{\alpha}_{2}$, then $G\left(\right)open="("\; close=")">{\alpha}_{1}.$

**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$${\nu}_{AL}\left(\right)open="("\; close=")">z;\kappa -{\kappa}^{2}\mathrm{exp}\left(\right)open="("\; close=")">-\frac{\sqrt{2}}{\kappa}z$$
- ${\nu}_{AL}\left(\right)open="("\; close=")">z;\kappa $ is an increasing negative function of z when $\kappa >1$, and it is a decreasing positive function of z when $0<\kappa <1$. ${\nu}_{AL,1}\left(\right)open="("\; close=")">z;1$, 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(\right)open="("\; close=")">z;\kappa $ but, when $\kappa \ne 1$, the function never reaches that limit value. To prove these results, it is sufficient to note that$$\frac{d{\nu}_{AL}\left(\right)open="("\; close=")">z;\kappa}{}dz-\mathrm{exp}\left(\right)open="("\; close=")">-\sqrt{2}\kappa z$$
- At $z=0$, the skewness function takes the following value:$${\nu}_{AL}\left(\right)open="("\; close=")">0;\kappa $$
- ${\nu}_{AL}\left(\right)open="("\; close=")">z;\kappa $ is a strictly decreasing function on $\kappa $. This is easily shown by means of$$\frac{d{\nu}_{AL}\left(\right)open="("\; close=")">z;\kappa}{}d\kappa \left(\right)open="["\; close="]">\mathrm{exp}\left(\right)open="("\; close=")">-\sqrt{2}\frac{z}{\kappa}$$

**Proposition**

**6.**

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

## 3. The Beta and the AST Distributions

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

## 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