# A Note on Ordering Probability Distributions by Skewness

^{1}

^{2}

^{*}

^{†}

## 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

## References

- Shaked, M.; Shanthikumar, J.G. Stochastic Orders. In Springer Series in Statistics 43; Springer: New York, NY, USA, 2007. [Google Scholar]
- Lehmann, E.L. Ordered families of distributions. Ann. Math. Statist.
**1955**, 26, 399–419. [Google Scholar] [CrossRef] - Arnold, B.C. Majorization and the Lorenz Order: A brief introduction. In Lecture Notes in Statistics 43; Springer: New York, NY, USA, 1987. [Google Scholar]
- Nanda, A.K.; Shaked, M. The hazard rate and reverse hazard rate orders, with applications to order statistics. Ann. Inst. Statist. Math.
**2001**, 53, 853–864. [Google Scholar] [CrossRef] - Ramos–Romero, H.M.; Sordo–Díaz, M.A. The proportional likelihood ratio order and applications. Questiio
**2001**, 25, 211–223. [Google Scholar] - Gupta, A.K.; Aziz, M.A.S. Convex Ordering of Random Variables and its Applications in Econometrics and Actuarial Science. Eur. J. Pure Appl. Math.
**2010**, 3, 79–85. [Google Scholar] - Oja, H. On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Stat.
**1981**, 8, 154–168. [Google Scholar] - Van Zwet, W.R. Mean, Median, Mode II. Stat. Neerl.
**1979**, 33, 1–5. [Google Scholar] [CrossRef] - MacGillivray, H.L. Skewness and asymmetry: measures and orderings. Ann. Stat.
**1986**, 14, 994–1011. [Google Scholar] [CrossRef] - Arnold, B.C.; Groeneveld, R.A. Measuring Skewness with respect to the Mode. Am. Stat.
**1995**, 49, 34–38. [Google Scholar] - Sato, M. Some remarks on the mean, median, mode and skewness. Aust. J. Stat.
**1997**, 39, 219–224. [Google Scholar] [CrossRef] - Von Hippel, P.T. Mean, Median and Skew: Correcting a Textbook Rule. J. Stat. Educ.
**2005**, 13. [Google Scholar] [CrossRef] - Das, S.; Mandal, P.K.; Ghosh, D. On Homogeneous Skewness of Unimodal Distributions. Indian J. Stat.
**2009**, 71-B, 187–205. [Google Scholar] - Rubio, F.J.; Steel, M. On the Marshall–Olkin transformation as a skewing mechanism. Comput. Stat. Data Anal.
**2012**, 56, 2251–2257. [Google Scholar] [CrossRef] [Green Version] - García, V.J.; Martel–Escobar, M.; Vázquez–Polo, F.J. Complementary information for skewness measures. Stat. Neerl.
**2015**, 69, 442–459. [Google Scholar] [CrossRef] - Mc Gill, W.J. Random fluctuations of response rate. Psychometrika
**1962**, 27, 3–17. [Google Scholar] [CrossRef] - Holla, M.S.; Bhattacharya, S.K. On a compound Gaussian distribution. Ann. Instit. Stat. Math.
**1968**, 20, 331–336. [Google Scholar] [CrossRef] - Kozubowski, T.J.; Podgórski, K. Maximum likelihood estimation of asymmetric Laplace parameters. Ann. Inst. Stat. Math.
**2002**, 54, 816–826. [Google Scholar] - Kotz, S.; van Dorp, J.R. Uneven two-sided power distribution: with applications in econometric models. Stat. Methods Appl.
**2004**, 13, 285–313. [Google Scholar] [CrossRef] - Fernández, C.; Steel, M.F.J. On Bayesian Modeling of Fat Tails and Skewness. J. Am. Stat. Assoc.
**1998**, 93, 359–371. [Google Scholar] - Johnson, D. The triangular distribution as a proxy for the beta distribution in risk analysis. The Statistician
**1997**, 46, 387–398. [Google Scholar] [CrossRef]

**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).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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