# Properties and Applications of a New Family of Skew Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Theorem 1.**

#### 2.1. The First Family of Skew Distributions

**Theorem**

**2.**

**Proof.**

**Proposition 1.**

- $\left(i\right)$
- ${f}_{Y}(0;\lambda ,\alpha )={g}_{Y}\left(0\right)$.
- $\left(ii\right)$
- ${f}_{Y}(y;\lambda ,\alpha )={f}_{Y}(y;\lambda ,-\alpha )$ for $\lambda \in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\alpha \in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}}-\left\{0\right\}$.
- $\left(iii\right)$
- ${f}_{Y}(y;0,\alpha )={g}_{Y}\left(y\right)$ for $\alpha \in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}}-\left\{0\right\}$.
- $\left(iv\right)$
- ${lim}_{\alpha \to 0}{f}_{Y}(y;\lambda ,\alpha )=2{g}_{Y}\left(y\right){G}_{X}\left(\lambda y\right)$ for $\lambda \in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}}$.
- $\left(v\right)$
- ${G}_{Y}(y;\lambda ,\alpha )=1-{G}_{Y}(-y;,-\lambda ,\alpha )$.
- $\left(vi\right)$
- If Y has the pdf (3) then $-Y$ has the same distribution but with the parameter λ replaced by $-\lambda $.
- $\left(vii\right)$
- ${f}_{Y}(y;\lambda ,\alpha )+{f}_{Y}(y;-\lambda ,\alpha )=2{g}_{Y}\left(y\right)$.
- $\left(viii\right)$
- Let $\alpha =\lambda $ in (3) and consider the two random variates ${Z}_{1}$ and ${Z}_{2}$ following the pdf (3) with parameters ${\lambda}_{1}\in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}}$ and ${\lambda}_{2}\in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathrm{R}\phantom{\rule{1.0pt}{0ex}}$, respectively, then, if ${\lambda}_{1}<{\lambda}_{2}$, ${Z}_{1}{<}_{st}{Z}_{2}$. That is, ${Z}_{1}$ is stochastically smaller than ${Z}_{2}$.

#### 2.2. The Second Family of Skew Distributions

**Theorem**

**3.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 3. The Normal Distribution Case

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

- $\left(i\right)$
- ${\mathsf{\Phi}}_{\lambda ,0}\left(z\right)=\mathsf{\Phi}\left(z\right)+T\left(z,\frac{\alpha}{z}-\lambda \right)-T\left(z,\frac{\alpha}{z}+\lambda \right)$.
- $\left(ii\right)$
- ${\mathsf{\Phi}}_{\lambda ,\alpha}\left(0\right)=\frac{1}{2}-2T\left(\alpha \delta ,\lambda \right)$.
- $\left(iii\right)$
- ${\mathsf{\Phi}}_{\lambda ,\alpha}\left(z\right)={\mathsf{\Phi}}_{-\lambda ,\alpha}(-z)$.

**Proof.**

**Proposition**

**6.**

**Proof.**

## 4. Estimation

## 5. Multivariate Versions

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**Theorem**

**5.**

**Proof.**

## 6. Numerical Illustrations

- $f\left(y;\mu ,\sigma ,\lambda ,\epsilon \right)=\left\{\begin{array}{cc}\frac{1}{2\sigma {c}_{\lambda}}\varphi \left(\frac{y-\mu}{\sigma (1+\epsilon )}-\lambda \right)\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{1.em}{0ex}}y<\mu \hfill \\ \frac{1}{2\sigma {c}_{\lambda}}\varphi \left(\frac{y-\mu}{\sigma (1-\epsilon )}+\lambda \right)\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{1.em}{0ex}}y\ge \mu \hfill \end{array}\right.$
- $f(y;\mu ,{\mu}_{1},\sigma ,{\sigma}_{1},p)=\frac{p}{\sigma}\phantom{\rule{0.166667em}{0ex}}\varphi \left(\frac{y-\mu}{\sigma}\right)+\frac{1-p}{{\sigma}_{1}}\phantom{\rule{0.166667em}{0ex}}\varphi \left(\frac{y-{\mu}_{1}}{{\sigma}_{1}}\right)$
- $f(y;\mu ,\sigma ,\lambda ,\alpha )=\frac{2}{\sigma}\varphi \left(\frac{y-\mu}{\sigma}\right)\mathsf{\Phi}\left(\lambda \left(\frac{y-\mu}{\sigma}\right)+\alpha {\left(\frac{y-\mu}{\sigma}\right)}^{3}\right)$

#### 6.1. Example 1

#### 6.2. Example 2

#### 6.3. Example 3

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Age | Number |
---|---|

1 | 1 |

5 | 89 |

10 | 342 |

15 | 718 |

20 | 2352 |

25 | 3593 |

30 | 3243 |

35 | 2533 |

40 | 2015 |

45 | 1747 |

50 | 1562 |

55 | 1662 |

60 | 1801 |

65 | 1915 |

70 | 1855 |

75 | 1611 |

80 | 1203 |

85 | 642 |

90 | 247 |

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**Figure 1.**Plot of the pdf (4) (thick line) denoted as GSN($\lambda ,\alpha $) and the SN($\lambda ,\mu ,\sigma $) (thin line) for selected values of the parameters $\alpha $ and $\lambda $.

**Figure 2.**Plot of the pdf in (10) for selected values of parameters and comparison with the skew normal one.

**Figure 3.**Density for BGSN for: ${\lambda}_{1}={\lambda}_{2}=-3$ and $\alpha =1$ (

**upper left**panel); ${\lambda}_{1}={\lambda}_{2}=3$ and $\alpha =1$ (

**upper right**panel); ${\lambda}_{1}={\lambda}_{2}=-3$ and $\alpha =5$ (

**lower left**panel); ${\lambda}_{1}={\lambda}_{2}=3$ and $\alpha =5$ (

**lower right**panel).

**Figure 6.**FSN distribution (dashed line) and GSN distribution (solid line) for the Kaposi’s sarcoma data.

$\mathit{\lambda}$ | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

Mean $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.564 | 0.713 | 0.756 | 0.774 | 0.782 |

Mean $\mathrm{GSN}(\lambda ,0.1)$ | 0.000 | 0.562 | 0.712 | 0.756 | 0.773 | 0.782 |

Variance $\mathrm{SN}\left(\lambda \right)$ | 1.000 | 0.681 | 0.490 | 0.427 | 0.400 | 0.387 |

Variance $\mathrm{GSN}(\lambda ,0.1)$ | 1.000 | 0.683 | 0.491 | 0.427 | 0.401 | 0.388 |

Mean $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.564 | 0.713 | 0.756 | 0.774 | 0.782 |

Mean $\mathrm{GSN}(\lambda ,1)$ | 0.000 | 0.439 | 0.645 | 0.720 | 0.751 | 0.767 |

Variance $\mathrm{SN}\left(\lambda \right)$ | 1.000 | 0.681 | 0.490 | 0.427 | 0.400 | 0.387 |

Variance $\mathrm{GSN}(\lambda ,1)$ | 1.000 | 0.806 | 0.583 | 0.481 | 0.435 | 0.410 |

Mean $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.564 | 0.713 | 0.756 | 0.774 | 0.782 |

Mean $\mathrm{GSN}(\lambda ,5)$ | 0.000 | 0.001 | 0.058 | 0.216 | 0.371 | 0.483 |

Variance $\mathrm{SN}\left(\lambda \right)$ | 1.000 | 0.681 | 0.490 | 0.427 | 0.400 | 0.387 |

Variance $\mathrm{GSN}(\lambda ,5)$ | 1.000 | 0.999 | 0.996 | 0.952 | 0.862 | 0.765 |

**Table 2.**Third and fourth standardized cumulants of the GSN and the SN variates for some parameter values.

$\mathit{\lambda}$ | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${\gamma}_{1}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.136 | 0.453 | 0.667 | 0.784 | 0.850 |

${\gamma}_{1}$, Mean $\mathrm{GSN}(\lambda ,0.1)$ | 0.000 | 0.135 | 0.451 | 0.664 | 0.782 | 0.849 |

${\gamma}_{2}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.061 | 0.305 | 0.509 | 0.632 | 0.705 |

${\gamma}_{2}$, $\mathrm{GSN}(\lambda ,0.1)$ | 0.000 | 0.060 | 0.302 | 0.507 | 0.630 | 0.703 |

${\gamma}_{1}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.136 | 0.453 | 0.667 | 0.784 | 0.850 |

${\gamma}_{1}$, $\mathrm{GSN}(\lambda ,1)$ | 0.000 | 0.082 | 0.281 | 0.488 | 0.639 | 0.738 |

${\gamma}_{2}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.061 | 0.305 | 0.509 | 0.632 | 0.705 |

${\gamma}_{2}$, $\mathrm{GSN}(\lambda ,1)$ | 0.000 | −0.046 | 0.071 | 0.289 | 0.458 | 0.571 |

${\gamma}_{1}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.136 | 0.453 | 0.667 | 0.784 | 0.850 |

${\gamma}_{1}$, $\mathrm{GSN}(\lambda ,5)$ | 0.000 | 0.006 | 0.188 | 0.336 | 0.332 | 0.311 |

${\gamma}_{2}$, $\mathrm{SN}\left(\lambda \right)$ | 0.000 | 0.061 | 0.305 | 0.509 | 0.632 | 0.705 |

${\gamma}_{2}$, $\mathrm{GSN}(\lambda ,5)$ | 0.000 | −0.000 | −0.044 | −0.294 | −0.480 | −0.501 |

n | $\overline{\mathit{y}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

315 | 12.789 | 28.411 | 1.147 | 5.425 |

Parameter | FESN | GSN |
---|---|---|

$\mu $ | 7.176 (0.405) | 6.714 (0.329) |

$\sigma $ | 4.396 (0.446) | 8.076 (0.406) |

$\lambda $ | −0.288 (0.235) | 9.692 (2.510) |

$\alpha $ | - | 2.486 (0.764) |

$\epsilon $ | −0.695 (0.048) | - |

${\ell}_{max}$ | −949.458 | −945.575 |

AIC | 1906.916 | 1899.150 |

CAIC | 1925.926 | 1918.160 |

n | $\overline{\mathit{y}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

240 | 3.276 | 1.964 | 0.882 | 5.049 |

Parameter | MN | GSN |
---|---|---|

$\mu $ | 2.115 (0.115) | 2.577 (0.126) |

$\sigma $ | 0.498 (0.104) | 1.564 (0.091) |

$\lambda $ | - | 3.780 (0.852) |

$\alpha $ | - | 3.685 (0.944) |

${\mu}_{1}$ | 3.727 (0.176) | - |

${\sigma}_{1}$ | 1.376 (0.084) | - |

p | 0.280 (0.077) | - |

AIC | 828.214 | 826.585 |

CAIC | 850.617 | 844.507 |

n | $\overline{\mathit{y}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

29131 | 45.396 | 416.487 | 0.313 | 1.936 |

Parameter | FSN | GSN |
---|---|---|

$\mu $ | 17.896 (0.119) | 37.039 (0.139) |

$\sigma $ | 34.245 (0.171) | 22.052 (0.105) |

$\lambda $ | 6.016 (0.118) | 4.898 (0.118) |

$\alpha $ | −1.007 (0.090) | 5.525 (0.138) |

AIC | 255,356.2 | 253,832.6 |

CAIC | 255,393.3 | 253,869.7 |

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**MDPI and ACS Style**

Gómez-Déniz, E.; Arnold, B.C.; Sarabia, J.M.; Gómez, H.W. Properties and Applications of a New Family of Skew Distributions. *Mathematics* **2021**, *9*, 87.
https://doi.org/10.3390/math9010087

**AMA Style**

Gómez-Déniz E, Arnold BC, Sarabia JM, Gómez HW. Properties and Applications of a New Family of Skew Distributions. *Mathematics*. 2021; 9(1):87.
https://doi.org/10.3390/math9010087

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Barry C. Arnold, José M. Sarabia, and Héctor W. Gómez. 2021. "Properties and Applications of a New Family of Skew Distributions" *Mathematics* 9, no. 1: 87.
https://doi.org/10.3390/math9010087