# The Asymmetric Alpha-Power Skew-t Distribution

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Alpha-Power Skew-t Distribution

**Definition**

**1.**

**Proposition**

**1.**

- (i)
- if $\lambda =0$, then $X\sim \mathrm{PT}(\alpha ,\nu )$,
- (ii)
- if $\alpha =1$, then $X\sim \mathrm{ST}(\lambda ,\nu )$,
- (iii)
- if $\lambda =0$ and $\alpha =1$, then $X\sim \mathrm{T}(\nu )$, where $\mathrm{T}(\nu )$ denotes the Student-t disribution with ν degree of freedom.
- (iv)
- if $\nu \to +\infty $, then $X\sim \mathrm{APSN}(\lambda ,\alpha )$,
- (v)
- if $\lambda =0$ and $\nu \to +\infty $, then $X\sim \mathrm{PN}(\alpha )$,
- (vi)
- if $\alpha =1$ and $\nu \to +\infty $, then $X\sim \mathrm{SN}(\lambda )$,
- (vii)
- if $\lambda =0$, $\alpha =1$ and $\nu \to +\infty $, then $X\sim \mathrm{N}(0,1)$,

**Proof.**

#### 2.1. Moments

**Proposition**

**2.**

**Proof.**

#### 2.2. Distribution Function

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

#### 2.3. Location and Scale Extension

**Definition**

**2.**

## 3. Statistical Inference for APST Distribution

**I**. Therefore, we use the observed information matrix to calculate the standard errors in the rest of the document.

#### 3.1. Extension to Censored Data

**Definition**

**3.**

#### 3.2. Properties of the CAPST Model

- If $\alpha =1$, then $Y\sim \mathrm{CST}(\mu ,\sigma ,\lambda ,\nu )$, where CST indicates the censored skew-t model.
- If $\lambda =0$, then $Y\sim \mathrm{CPT}(\mu ,\sigma ,\alpha ,\nu )$, where CPT indicates the censored power-t model.
- If $\alpha =1$ and $\lambda =0$, then $Y\sim \mathrm{CT}(\mu ,\sigma ,\nu )$, that is, the censored Student-t model follows.
- If $\nu \to +\infty $, then $Y\sim \mathrm{CAPSN}(\mu ,\sigma ,\lambda ,\alpha )$, where CAPSN indicates the censored alpha-power skew-normal model.
- If $\alpha =1$ and $\nu \to +\infty $, then $Y\sim \mathrm{CSN}(\mu ,\sigma ,\lambda )$, that is, the censored skew-normal model follows.
- If $\lambda =0$ and $\nu \to +\infty $, then $Y\sim \mathrm{CPN}(\mu ,\sigma ,\alpha )$, that is, the censored power-normal model follows.
- If $\alpha =1$, $\lambda =0$ and $\nu \to +\infty $, then $Y\sim \mathrm{CN}(\mu ,{\sigma}^{2})$, that is, the censored normal model follows.

## 4. Real Data Applications

#### 4.1. Application 1: Volcano Heights Data

#### 4.2. Application 2: Stellar Abundances Data

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Probability density function of $\mathrm{APST}(\lambda ,\alpha ,10)$ for some values of $\lambda $ and $\alpha $.

**Figure 3.**(

**Left**) Graph of fitted densities to volcano height data. (

**Right**) Empirical CDF and CDF of fitted APST model.

**Figure 4.**Stellar abundances data. Envelopes of transformed martingale residuals for CT, CST, CPT, and CAPST models.

**Table 1.**Skewness and kurtosis for the models $\mathrm{ST}(\lambda ,\nu )$, $\mathrm{PT}(\alpha ,\nu )$, and $\mathrm{APST}(\lambda ,\alpha ,\nu )$, for $\lambda \in (-40,40)$, $\alpha \in (0.5,50)$ and $\nu =2,\cdots 7$.

Skew$\u2014\mathit{t}$ | Power$\u2014\mathit{t}$ | Alpha—Power Skew $\u2014\mathit{t}$ | ||||
---|---|---|---|---|---|---|

$\mathbf{\nu}$ | Skewness | Kurtosis | Skewness | Kurtosis | Skewness | Kurtosis |

2 | $(-0.963,0.963)$ | $(3.170,3.489)$ | $(-0.119,3.040)$ | $(1.552,10.436)$ | $(-2.452,14.314)$ | $(1.395,864.385)$ |

3 | $(-0.950,0.950)$ | $(3.146,3.357)$ | $(-0.086,1.362)$ | $(1.325,3.223)$ | $(-2.130,4.902)$ | $(1.628,114.098)$ |

4 | $(-1.853,1.853)$ | $(5.099,7.824)$ | $(-0.530,1.178)$ | $(3.461,5.299)$ | $(-1.898,3.215)$ | $(3.153,29.874)$ |

5 | $(-0.947,0.947)$ | $(3.051,3.327)$ | $(-0.475,0.271)$ | $(1.176,3.130)$ | $(-1.968,3.046)$ | $(3.862,19.925)$ |

6 | $(-1.681,1.681)$ | $(4.554,7.279)$ | $(-0.533,1.118)$ | $(3.974,5.173)$ | $(-1.681,2.145)$ | $(3.892,11.893)$ |

7 | $(-0.944,0.944)$ | $(3.007,3.367)$ | $(-0.710,0.243)$ | $(1.264,3.082)$ | $(-1.535,2.536)$ | $(3.136,15.924)$ |

n | Mean | Variance | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

1520 | 16.7760 | 15.6682 | 0.6461 | 4.3809 |

Distribution | ||||
---|---|---|---|---|

Estimates | Student-t | ST | PT | APST |

$\widehat{\mu}$ | 14.7835(0.3615 ) | 4.7469(0.6892) | 8.4027(0.7923) | 11.5509(0.1337) |

$\widehat{\sigma}$ | 11.0045(0.3975) | 14.1532(0.7237) | 11.8146(0.4707) | 22.6885(0.0792) |

$\widehat{\lambda}$ | – | 1.5673(0.1838) | – | 5.2347(0.2870) |

$\widehat{\alpha}$ | – | – | 1.7912(0.1147) | 0.3205(0.0347) |

$\widehat{\nu}$ | 3.4156(0.3601) | 3.4075(0.3454) | 2.7473(0.2566) | 12.8734(2.9729) |

$\widehat{\ell}$ | −6273.35 | −6219.25 | −6228.77 | −6205.94 |

AIC | 12,552.70 | 12,446.49 | 12,465.53 | 12,421.87 |

BIC | 12,568.68 | 12,467.79 | 12,486.53 | 12,448.50 |

CAIC | 12,571.68 | 12,471.79 | 12,490.83 | 12,453.50 |

Distribution | ||||
---|---|---|---|---|

Estimates | CT | CST | CPT | CAPST |

$\widehat{\mu}$ | 1.0314(0.0010) | 1.2306(0.0018) | 1.2098(0.0052) | 1.1761(0.0054) |

$\widehat{\sigma}$ | 0.1596(0.0012) | 0.2712(0.0058) | 0.0818(0.0008) | 0.0905(0.0020) |

$\widehat{\lambda}$ | – | −3.5655(3.7748) | – | 0.6580(0.5031) |

$\widehat{\alpha}$ | – | – | 0.1705(0.0208) | 0.1518(0.0251) |

$\widehat{\nu}$ | 0.9974(0.0884) | 1.2501(0.1774) | 6.0927(0.7501) | 6.0999(0.7326) |

$\widehat{\ell}$ | −29.50743 | −18.87016 | −17.67113 | −14.80241 |

AIC | 65.01487 | 45.74033 | 43.34227 | 39.60482 |

BIC | 71.67339 | 54.61836 | 52.22030 | 50.70236 |

CAIC | 59.38987 | 38.37525 | 35.97719 | 30.57256 |

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**MDPI and ACS Style**

Tovar-Falón, R.; Bolfarine, H.; Martínez-Flórez, G.
The Asymmetric Alpha-Power Skew-*t* Distribution. *Symmetry* **2020**, *12*, 82.
https://doi.org/10.3390/sym12010082

**AMA Style**

Tovar-Falón R, Bolfarine H, Martínez-Flórez G.
The Asymmetric Alpha-Power Skew-*t* Distribution. *Symmetry*. 2020; 12(1):82.
https://doi.org/10.3390/sym12010082

**Chicago/Turabian Style**

Tovar-Falón, Roger, Heleno Bolfarine, and Guillermo Martínez-Flórez.
2020. "The Asymmetric Alpha-Power Skew-*t* Distribution" *Symmetry* 12, no. 1: 82.
https://doi.org/10.3390/sym12010082