# Univariate and Bivariate Models Related to the Generalized Epsilon–Skew–Cauchy Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Representation of the ESC (Epsilon–Skew–Cauchy) Model

**Proposition**

**1.**

**Proof.**

## 3. Generalized Cauchy Distribution

**Proposition**

**2.**

**Proof.**

- If ${\epsilon}_{2}=0$ a generalized Cauchy (GC) distribution is obtained. In this case we write $Z\sim GC\left(\epsilon \right)$, and its pdf is given by$${f}_{Z}\left(\right)open="("\; close=")">z;\epsilon $$
- If ${\epsilon}_{1}={\epsilon}_{2}=\epsilon $ an epsilon–skew–Cauchy distribution is obtained. In this case we write $Z\sim ESC2\left(\epsilon \right)$, and its pdf is given by$${f}_{Z}\left(\right)open="("\; close=")">z;\epsilon $$

## 4. General Bivariate Mudholkar-Hutson Distribution

- Mudholkar and Hutson type. For this we set: ${\alpha}_{i}=1+{\epsilon}_{i}$, ${\beta}_{i}=1-{\epsilon}_{i}$ and ${\gamma}_{i}=\left(\right)open="("\; close=")">1+{\epsilon}_{i}$ for $i=1,2,3$.In this case the density (4) simplifies somewhat to become:$$\begin{array}{ccc}f(x,y)& =& \frac{1}{4\pi}\left(\right)open="\{"\; close="\}">{(1+{\epsilon}_{3})}^{3}{\left(\right)}^{1}+{\left(\right)}^{\frac{1+{\epsilon}_{3}}{1+{\epsilon}_{2}}}2\hfill & -3/2\\ I(x0,y0)\end{array}$$A further specialization of the density (5) can be considered, as follows.
- Homogenous Mudholkar and Hutson type. For this we set: ${\alpha}_{i}=1+\epsilon $, ${\beta}_{i}=1-\epsilon $ and ${\gamma}_{i}=\left(\right)open="("\; close=")">1+\epsilon $ for $i=1,2,3$.This homogeneity results in a little simplification of (4), thus:$$\begin{array}{c}f\left(\right)open="("\; close=")">x,y;\epsilon \\ =& \frac{1}{4\pi}\left(\right)open="\{"\; close="\}">\left(\right)open="["\; close="]">{(1+\epsilon )}^{3}+{(1-\epsilon )}^{3}{\left(\right)}^{1}-3/2\hfill & I(x0,y0)\end{array}$$It is easy to see that the parameter $\epsilon $ is not identifiable in (6) because $f\left(\right)open="("\; close=")">x,y;\epsilon $. An adjustment to ensure identifiability involves introducing the constraint $\epsilon \ge 0$.
- Equal weights. In this case we assume that ${\alpha}_{1},$ ${\alpha}_{2},$ ${\alpha}_{3},$ ${\beta}_{1},{\beta}_{2},{\beta}_{3}$ are positive numbers and ${\gamma}_{i}=1/2$ for $i=1,2,3$.$$\begin{array}{c}f\left(\right)open="("\; close=")">x,y\\ =& \frac{1}{4\pi}\left(\right)open="\{"\; close="\}">\frac{{\alpha}_{3}^{2}}{{\alpha}_{1}{\alpha}_{2}}{\left(\right)}^{1}+{\left(\right)}^{\frac{{\alpha}_{3}}{{\alpha}_{2}}}2\hfill & -3/2\end{array}$$The pdf (7) is not identifiable because the values of ${\alpha}_{i}$ can be interchanged with those of ${\beta}_{i}$ and $f(x,y)$ does not change. Moreover, multiplying all of the $\alpha $’s and $\beta $’s by a constant does not change $f(x,y)$. So, one way to get identifiability in the model (7) is to set ${\alpha}_{i}={\beta}_{i}$ $\left(\right)$ and ${\alpha}_{3}$ equal to 1. In that case, Equation (7) takes the form$$f\left(\right)open="("\; close=")">x,y$$However, this is then recognizable as being simply a scaled version of the standard bivariate Cauchy density (compare with Equation (3)).

**Proposition**

**3.**

**Proof.**

## 5. Application

- When the pdf is given by Equation (4). That is taking $\nu \to \infty $ in the bivariate skew t MH model.
- When the pdf is given by$$f\left(\right)open="("\; close=")">x,y;{\epsilon}_{i},{\mu}_{j},{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12},$$
- When the pdf is given by$$f\left(\right)open="("\; close=")">x,y;\epsilon ,{\mu}_{j},{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12},$$
- When the pdf is given by$$f\left(\right)open="("\; close=")">x,y;{\mu}_{j},{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12},$$

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples of the $ESC\left(\epsilon \right)$ density for: $\epsilon =0$ (green line), $\epsilon =-0.5$ (blue line), $\epsilon =-0.8$ (black line), $\epsilon =0.5$ (red line) and $\epsilon =0.8$ (pink line).

**Figure 4.**Australian athletes data: scatter plot (Ht, Wt) and fitted General bivariate MH. Red point for Men and sign + for women.

Model | AIC | $\left(\right)open="("\; close=")">{\mathit{\alpha}}_{\mathit{i}},{\mathit{\beta}}_{\mathit{i}},{\mathit{\gamma}}_{\mathit{i}},\mathit{\mu},\mathit{\Sigma}$ Estimates | Bootstrap Standard Errors of the Estimates |
---|---|---|---|

Bivariate skew t MH | $1354.5$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">13.48,16.62,1.25\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">10.65,36.47,3.00\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">10.02,29.96,8.05\hfill \end{array}\left(\right)open="("\; close=")">0.006,0.02,0.03\hfill $ |

General bivariate MH | $1413.5$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">13.26,16.04,1.05\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">10.64,36.50,4.99\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">2.70,4.30,0.90\hfill \end{array}\left(\right)open="("\; close=")">0.003,0.02,0.02\hfill $ |

Bivariate MH (1.) | $1429.4$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\epsilon}_{1},{\epsilon}_{2},{\epsilon}_{3}=\left(\right)open="("\; close=")">0.85,0.83,-1.00\hfill \end{array}\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">186.30,83.80\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">0.07,0.08,0.003\hfill \end{array}\left(\right)open="("\; close=")">33.30,51.34,36.83\hfill $ |

Bivariate MH (2.) | $1453.8$ | $\begin{array}{c}\epsilon =0.50\hfill \\ \left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">171.60,62.30\hfill \end{array}\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}=\left(\right)open="("\; close=")">32.49,61.40,33.96\hfill $ | $\begin{array}{c}0.15\hfill \\ \left(\right)open="("\; close=")">1.86,2.80\hfill \end{array}$ |

Bivariate Cauchy | $1665.5$ | $\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}$ $\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}$ | $\left(\right)$ $\left(\right)$ |

Model | AIC | $\left(\right)open="("\; close=")">{\mathit{\alpha}}_{\mathit{i}},{\mathit{\beta}}_{\mathit{i}},{\mathit{\gamma}}_{\mathit{i}},\mathit{\mu},\mathit{\Sigma}$ Estimates | Bootstrap Standard Errors of the Estimates |
---|---|---|---|

Bivariate skew t MH | $1396.9$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">13.43,20.16,2.62\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">35.02,11.84,6.01\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">14.66,5.34,4.12\hfill \end{array}\left(\right)open="("\; close=")">0.02,0.02,0.04\hfill $ |

General bivariate MH | $1441.8$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">52.54,52.17,14.58\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">29.83,62.76,31.76\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">0.89,1.28,0.43\hfill \end{array}\left(\right)open="("\; close=")">0.01,0.01,0.02\hfill $ |

Bivariate MH (1.) | $1456.2$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\epsilon}_{1},{\epsilon}_{2},{\epsilon}_{3}=\left(\right)open="("\; close=")">-0.87,-0.91,-1.0\hfill \end{array}\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">172.70,61.00\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">0.06,0.05,0.00\hfill \end{array}\left(\right)open="("\; close=")">32.88,62.62,38.52\hfill $ |

Bivariate MH (2.) | $1512.6$ | $\begin{array}{c}\epsilon =0.17\hfill \\ \left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">185.10,80.54\hfill \end{array}\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}=\left(\right)open="("\; close=")">36.15,70.27,38.53\hfill $ | $\begin{array}{c}0.21\hfill \\ \left(\right)open="("\; close=")">2.73,3.54\hfill \end{array}$ |

Bivariate Cauchy | $1775.2$ | $\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}$ $\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}$ | $\left(\right)$ $\left(\right)$ |

Model | AIC | $\left(\right)open="("\; close=")">{\mathit{\alpha}}_{\mathit{i}},{\mathit{\beta}}_{\mathit{i}},{\mathit{\gamma}}_{\mathit{i}},\mathit{\mu},\mathit{\Sigma}$ Estimates | Bootstrap Standard Errors of the Estimates |
---|---|---|---|

Bivariate skew t MH | $2803.3$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">14.15,17.35,1.09\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">49.35,67.05,1.62\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">8.18,10.35,3.57\hfill \end{array}\left(\right)open="("\; close=")">0.01,0.01,0.02\hfill $ |

General bivariate MH | $2901.4$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}=\left(\right)open="("\; close=")">15.42,17.67,2.25\hfill \end{array}\left(\right)open="("\; close=")">{\beta}_{1},{\beta}_{2},{\beta}_{3}=\left(\right)open="("\; close=")">49.28,66.53,3.36\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">10.74,12.92,2.61\hfill \end{array}\left(\right)open="("\; close=")">0.01,0.02,0.02\hfill $ |

Bivariate MH (1.) | $2984.0$ | $\begin{array}{c}\left(\right)open="("\; close=")">{\epsilon}_{1},{\epsilon}_{2},{\epsilon}_{3}=\left(\right)open="("\; close=")">1.00,0.39,-0.13\hfill \end{array}\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">181.06,74.48\hfill $ | $\begin{array}{c}\left(\right)open="("\; close=")">0.01,0.07,0.06\hfill \end{array}\left(\right)open="("\; close=")">1.59,5.00,2.17\hfill $ |

Bivariate MH (2.) | $3022.2$ | $\begin{array}{c}\epsilon =0.70\hfill \\ \left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}=\left(\right)open="("\; close=")">171.37,61.32\hfill \end{array}\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}=\left(\right)open="("\; close=")">48.62,103.69,64.24\hfill $ | $\begin{array}{c}0.10\hfill \\ \left(\right)open="("\; close=")">2.42,3.07\hfill \end{array}$ |

Bivariate Cauchy | $3536.9$ | $\left(\right)open="("\; close=")">{\mu}_{1},{\mu}_{2}$ $\left(\right)open="("\; close=")">{\Sigma}_{11},{\Sigma}_{22},{\Sigma}_{12}$ | $\left(\right)$ $\left(\right)$ |

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**MDPI and ACS Style**

Arnold, B.C.; Gómez, H.W.; Varela, H.; Vidal, I.
Univariate and Bivariate Models Related to the Generalized Epsilon–Skew–Cauchy Distribution. *Symmetry* **2019**, *11*, 794.
https://doi.org/10.3390/sym11060794

**AMA Style**

Arnold BC, Gómez HW, Varela H, Vidal I.
Univariate and Bivariate Models Related to the Generalized Epsilon–Skew–Cauchy Distribution. *Symmetry*. 2019; 11(6):794.
https://doi.org/10.3390/sym11060794

**Chicago/Turabian Style**

Arnold, Barry C., Héctor W. Gómez, Héctor Varela, and Ignacio Vidal.
2019. "Univariate and Bivariate Models Related to the Generalized Epsilon–Skew–Cauchy Distribution" *Symmetry* 11, no. 6: 794.
https://doi.org/10.3390/sym11060794