# The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model

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## Abstract

**:**

## 1. Introduction

## 2. Skew-Elliptical Alpha-Power Model

#### 2.1. Skew-Elliptical Sinh-Alpha-Power Model

- sinh-normal model, when $\lambda =0$ and $\alpha =1,$
- skew sinh-normal distribution, when $\alpha =1$
- sinh-normal power-normal distribution, when $\lambda =0$

**Theorem**

**1.**

**Theorem**

**2.**

- 1.
- $bT\sim PESNBS(\gamma ,b\tau ,\lambda ,\alpha ),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}b>0$ and
- 2.
- ${T}^{-1}\sim PESNBS(\gamma ,{\tau}^{-1},-\lambda ,\alpha ).$

#### 2.2. More Properties

- Let $Y\sim SPESN(\gamma ,\xi ,\eta ,\lambda ,\alpha )$; then, for constants $a\in \mathbb{R}$ and $b\in {\mathbb{R}}^{+},$$$W=a+bY\sim SPESN(\gamma ,a+b\xi ,b\eta ,\lambda ,\alpha ).$$In particular,$$W=bY\sim SPESN(\gamma ,b\xi ,b\eta ,\lambda ,\alpha ).$$
- Let $Y\sim SAPSE(\gamma ,\xi ,\eta ;g,\lambda ,\alpha )$; then,$$W=\frac{2}{\gamma}sinh\left(\right)open="("\; close=")">\frac{Y-\xi}{\eta}$$
- Let $Y\sim SPESN(\gamma ,\xi ,\eta ,\lambda ,\alpha )$; then,$$W=\frac{2}{\gamma}sinh\left(\right)open="("\; close=")">\frac{Y-\xi}{\eta}$$
- Let $Y\sim SPESN(\gamma ,\xi ,\eta ,\lambda ,1)$; then,$${W}^{2}=\frac{4}{{\gamma}^{2}}{sinh}^{2}\left(\right)open="("\; close=")">\frac{Y-\xi}{\eta}$$

#### 2.3. The Skew-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model

## 3. Inference

#### Observed Information Matrix

## 4. Simulation Study

## 5. Applications

#### 5.1. Application 1: Times to Failure

#### 5.2. Application 2: Patients with Advanced Lung Cancer Status

## 6. Conclusions

- The introduction of two extra shape parameters makes the error distribution more flexible, allowing it to incorporate additional kurtosis and asymmetry.
- Maximum likelihood properties of large samples such as consistency and asymptotic normality were established.
- A simulation study was performed to evaluate of the ML estimations. As expected, the bias and standard deviation of our estimation decrease as the sample size n increases.
- In the applications, AIC criteria statistics are used. These criteria indicate that the model that best fit the data is SPESN model.
- A Bayesian approach will be worked on in a future work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution PESN. (

**a**) $\alpha =1.5$ and $\lambda =-0.75$ (dotted dashed line), 0 (dotted line), 1 (dashed line) and 1.75 (solid line), (

**b**) $\lambda =0.70$ and $\alpha =$ 0.50 (dotted-dashed line), 1.0 (dotted line), 2.0 (dashed line), and 5.0 (solid line).

**Figure 2.**Plots for pdf ${\phi}_{SPESN}(y;\gamma ,\phantom{\rule{0.166667em}{0ex}}\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha ).$ (

**a**) $(\gamma ,\phantom{\rule{0.166667em}{0ex}}\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ (0.75, 0,2, 1.75, 1.5) (dashed and dotted line), (1.25, 0, 2, 1.75, 1.5) (dotted line), (1.75, 0, 2, 1.75, 1.5) (dashed line) and (2.25, 0, 2, 1.75, 1.5) (solid line). (

**b**) $(\gamma ,\phantom{\rule{0.166667em}{0ex}}\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ (1.75, 0, 2, −1.5, 1.5) (dashed and dotted line), $(1.75,0,2,-0.75,1.5)$ (dotted line), $(1.75,0,2,0.75,1.5)$ (dashed line) and $(1.75,0,2,1.5,1.5)$ (solid line). (

**c**) $(\gamma ,\phantom{\rule{0.166667em}{0ex}}\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ (0.75, 0, 2, 1.75, 0.75) (dashed and dotted line), (0.75, 0, 2, 1.75, 1.0) (dotted line), (0.75, 0, 2, 1.75, 1.25) (dashed line) and (0.75, 0, 2, 1.75, 1.5) (solid line).

**Figure 3.**Plots for density function ${\phi}_{SPESN}(y;\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha ).$ (

**a**) $(\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ (5.5, 0, 2, 0.5, 0.75) (dashed and dotted line), (7.5, 0, 2, 0.5, 0.75) (dotted line), (9.5, 0, 2, 0.5, 0.75) (dashed line), and (11.5, 0, 2, 0.5, 0.75) (solid line). (

**b**) $(\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ (5.5, 0, 2, −0.5, 0.75) (dashed and dotted line), (7.5, 0, 2, −0.5, 0.75) (dotted line), (9.5, 0, 2, −0.5, 0.75) (dashed line), and (11.5, 0, 2, −0.5, 0.75) (solid line).

**Figure 4.**Plots for $r\left(t\right)$ (

**a**) $(\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ $(0.5,0,2,1.75,1.5)$ (dashed and dotted line), $(0.75,0,2,1.75,1.5)$ (dotted line), $(1.25,0,2,1.75,1.5)$ (dashed line), and $(1.75,0,2,1.75,1.5)$ (solid line). (

**b**) $(\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ $(1.75,1,2,-1.5,1.5)$ (dashed and dotted line), $(1.75,1,2,-0.75,1.5)$ (dotted line), $(1.75,1,2,0.75,1.5)$ (dashed line), and (1.75,1,2,1.5,1.5) (solid line). (

**c**) $(\gamma ,\xi ,\phantom{\rule{0.166667em}{0ex}}\eta ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\alpha )=$ $(0.75,1,2,1.75,0.75)$ (dashed and dotted line), $(0.75,1,2,1.75,1.0)$ (dotted line), $(0.75,1,2,1.75,1.25)$ (dashed line), and $(0.75,1,2,1.75,1.5)$ (solid line).

**Figure 5.**Histogram for the scaled residuals Z, from the fitted models. (

**a**) SHN, (

**b**) SESN and (

**c**) SPESN.

**Figure 7.**Histogram for the scaled residuals Z, from the fitted models. (

**a**) SHN, (

**b**) SESN and (

**c**) SPESN.

True | n = 100 | n = 200 | n = 500 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{\beta}}_{\mathbf{0}}$ | ${\mathbf{\beta}}_{\mathbf{1}}$ | $\mathbf{\gamma}$ | $\mathbf{\lambda}$ | $\mathbf{\alpha}$ | Bias | se | MSE | Bias | se | MSE | Bias | se | MSE | |

−0.5 | −0.75 | 0.5 | −0.9 | 1.3 | ${\beta}_{0}$ | 0.0921 | 0.5944 | 0.6525 | 0.0543 | 0.3668 | 0.4003 | 0.0352 | 0.2053 | 0.2334 |

${\beta}_{1}$ | −0.0766 | 0.7066 | 0.7719 | −0.0679 | 0.5287 | 0.5875 | −0.0434 | 0.2037 | 0.2270 | |||||

$\gamma $ | 0.1795 | 1.7074 | 1.8050 | 0.0959 | 0.8442 | 0.9009 | 0.0685 | 0.6270 | 0.6635 | |||||

$\lambda $ | −0.1949 | 1.3205 | 1.4016 | −0.0454 | 1.2749 | 1.3221 | −0.0257 | 0.3577 | 0.3902 | |||||

$\alpha $ | 0.5088 | 1.6844 | 1.7630 | 0.2800 | 1.0306 | 1.0900 | 0.0609 | 0.3562 | 0.3795 | |||||

−1 | −1 | 0.25 | −0.3 | 1.5 | ${\beta}_{0}$ | −0.1087 | 0.6995 | 0.7749 | −0.0834 | 0.5313 | 0.5815 | −0.0649 | 0.3074 | 0.3320 |

${\beta}_{1}$ | 0.0980 | 0.7404 | 0.8209 | 0.0688 | 0.5500 | 0.5982 | 0.0505 | 0.3063 | 0.3290 | |||||

$\gamma $ | 0.1136 | 1.1513 | 1.2464 | 0.0766 | 1.0021 | 1.0450 | 0.0626 | 0.3697 | 0.3964 | |||||

$\lambda $ | −0.1217 | 1.5794 | 1.6623 | −0.0824 | 0.7816 | 0.8344 | −0.0437 | 0.4614 | 0.4854 | |||||

$\alpha $ | 0.4648 | 1.4808 | 1.5671 | 0.1977 | 0.5611 | 0.6067 | 0.0902 | 0.3467 | 0.3701 | |||||

1 | 1.5 | 0.75 | 0.3 | 2 | ${\beta}_{0}$ | 0.1089 | 0.6150 | 0.6655 | 0.0600 | 0.3760 | 0.4012 | 0.0529 | 0.2760 | 0.3021 |

${\beta}_{1}$ | −0.1019 | 0.5051 | 0.5774 | −0.0405 | 0.4160 | 0.4573 | −0.0257 | 0.2900 | 0.3186 | |||||

$\gamma $ | 0.1933 | 1.1773 | 1.2575 | 0.0659 | 1.0441 | 1.0983 | 0.0357 | 0.4767 | 0.5139 | |||||

$\lambda $ | 0.1595 | 1.6046 | 1.6647 | 0.0797 | 0.8889 | 0.9307 | 0.0448 | 0.5883 | 0.6182 | |||||

$\alpha $ | 0.3369 | 1.6177 | 1.7026 | 0.1618 | 1.2599 | 1.2910 | 0.0729 | 0.4479 | 0.4726 |

Parameters | SHN | SESN | SPESN |
---|---|---|---|

$\gamma $ | 1.279 (0.143) | 2.011 (0.313) | 5.379 (0.189) |

${\beta}_{0}$ | 0.097 (0.170) | −0.961 (0.166) | −2.620 (0.060) |

${\beta}_{0}^{*}$ | 0.165 | 0.289 | |

${\beta}_{1}$ | −14.116 (1.571) | −13.870 (1.602) | −13.602 (1.579) |

$\lambda $ | 1.642 (0.618) | −0.932 (0.174) | |

$\alpha $ | 13.889 (4.991) | ||

AIC | 129.235 | 125.360 | 122.917 |

Parameters | SHN | SESN | SPESN |
---|---|---|---|

$\gamma $ | 2.412 (0.153) | 6.963 ( 2.185) | 43.489 (19.792) |

${\beta}_{0}$ | 1.524 (0.837) | −1.356 ( 0.713) | −5.678 (0.954) |

${\beta}_{1}$ | 0.041 ( 0.015) | 0.034 (0.007) | 0.033 (0.004) |

${\beta}_{2}$ | −0.129 ( 0.509) | 0.488 ( 0.305) | 0.5776 (0.194) |

$\lambda $ | 8.505 (5.523) | 9.793 (16.669) | |

$\alpha $ | 2.468 (0.306) | ||

AIC | 322.208 | 193.285 | 154.786 |

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**MDPI and ACS Style**

Martínez-Flórez, G.; Bolfarine, H.; Gómez, Y.M.
The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model. *Symmetry* **2021**, *13*, 1297.
https://doi.org/10.3390/sym13071297

**AMA Style**

Martínez-Flórez G, Bolfarine H, Gómez YM.
The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model. *Symmetry*. 2021; 13(7):1297.
https://doi.org/10.3390/sym13071297

**Chicago/Turabian Style**

Martínez-Flórez, Guillermo, Heleno Bolfarine, and Yolanda M. Gómez.
2021. "The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model" *Symmetry* 13, no. 7: 1297.
https://doi.org/10.3390/sym13071297