# Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**(i)**we have $C(x,0)=C(0,y)=0$ for any $(x,y)\in {[0,1]}^{2}$,

**(ii)**we have $C(x,1)=x$ and $C(1,y)=y$ for any $(x,y)\in {[0,1]}^{2}$,

**(iii)**we have (in the absolutely continuous case)

- In [17], copulas of the following product form are investigated:$$\begin{array}{c}\hfill C(x,y)=xy\varphi \left[(1-x)(1-y)\right],\phantom{\rule{1.em}{0ex}}(x,y)\in {[0,1]}^{2},\end{array}$$
- One may also mention the original approach in [14], generating copulas of the following polyno-exponential form:$$\begin{array}{c}\hfill C(x,y)=xyexp\left[-\varphi \left(x\right)\psi \left(y\right)\right],\phantom{\rule{1.em}{0ex}}(x,y)\in {[0,1]}^{2},\end{array}$$

## 2. Ratio-Polynomial Copula

#### 2.1. Presentation

**Proposition**

**1.**

**Proof.**

**(i)**For any $x\in [0,1]$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**Using standard differentiation techniques and appropriate factorizations (hereafter, “appropriate” means “to choose a manageable one after a lot of possibilities tested”), for any $(x,y)\in {[0,1]}^{2}$, we have

**(iii)**is proved.

**Remark**

**1.**

#### 2.2. Related Functions and Copulas

#### 2.2.1. Main Functions

#### 2.2.2. Product of Copulas

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.3. Properties

**Lemma**

**3.**

**Proof.**

**Remark**

**2.**

**Lemma**

**4.**

**Proof.**

## 3. Second Ratio-Polynomial Copula

#### 3.1. Presentation

**Proposition**

**2.**

**Proof.**

**(i)**For any $x\in [0,1]$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**Using standard differentiation techniques and appropriate factorizations, for any $(x,y)\in {[0,1]}^{2}$, we have

**(iii)**is proved.

**Remark**

**3.**

#### 3.2. Properties

**Remark**

**4.**

**Lemma**

**5.**

**Proof.**

## 4. Ratio-Sine Copula

#### 4.1. Presentation

**Proposition**

**3.**

**Proof.**

**(i)**For any $x\in [0,1]$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**For any $(x,y)\in {[0,1]}^{2}$, using standard differentiation techniques and appropriate factorizations, we have

**(iii)**, we need to prove that, for any $u\in (0,\pi /2]$, $f\left(u\right)\ge 0$ and $g\left(u\right)\ge 0$. Let us begin with $g\left(u\right)$. The following inequality is well-known: $sin\left(u\right)\ge ucos\left(u\right)$ for any $u\in [0,\pi /2]$. Therefore, we have

**(iii)**is proved.

#### 4.2. Properties

**Remark**

**.**

## 5. Ratio-Arctangent Copula

#### 5.1. Presentation

**Proposition**

**4.**

**Proof.**

**(i)**For any $x\in [0,1]$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**For any $(x,y)\in {[0,1]}^{2}$, using standard differentiation techniques and appropriate factorizations, we have

**(iii)**, we need to prove that, for any $u>0$, $f\left(u\right)\ge 0$ and $g\left(u\right)\ge 0$.

**(iii)**is proved.

#### 5.2. Properties

**Remark**

**6.**

## 6. Ratio-Logarithmic Copula

#### 6.1. Presentation

**Proposition**

**5.**

**Proof.**

**(i)**For any $x\in [0,1]$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**For any $(x,y)\in {[0,1]}^{2}$, using standard differentiation techniques and appropriate factorizations, we have

**(iii)**, we need to prove that, for any $u>0$, $f\left(u\right)\ge 0$ and $g\left(u\right)\ge 0$.

**(iii)**is proved.

**(i)**and

**(ii)**of Definition 1 remain true), but the related admissible domain for the negative values conducted by the point

**(iii)**of Definition 1 remains a mathematical challenge. Numerical tests validate $\alpha \ge -1/2$ (a bit less in fact). We illustrate this conjecture by a plot of the RL copula for $\alpha =-1/2$ in Figure 19.

#### 6.2. Properties

**Remark**

**7.**

## 7. A Note on a Multi-Dimensional Approach

**Definition**

**2.**

**(i)**we have $C({x}_{1},\dots ,{x}_{i-1},0,{x}_{i+1},\dots ,{x}_{n})=0$ for any $({x}_{1},\dots ,{x}_{n})\in {[0,1]}^{n}$ and $i=1,\dots ,n$.

**(ii)**we have $C(1,\dots ,1,x,1,\dots ,1)=x$ for any $x\in [0,1]$, and this, in each of the n vector components (the n-dimensional function is equal to x if one vector component is x and all others are equal to 1).

**(iii)**we have

**Proposition**

**6.**

**Proof.**

**(i)**For any $({x}_{1},\dots ,{x}_{n})\in {[0,1]}^{n}$, we have

**(ii)**For any $x\in [0,1]$, we have

**(iii)**For any $({x}_{1},\dots ,{x}_{n})\in {[0,1]}^{n}$, using differentiation techniques and multiple (non-trivial) factorizations, we have

**(iii)**is proved.

**Remark**

**8.**

## 8. Summary

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 19.**Illustration of the conjecture: plots of the shapes and contours of the RL copula for $\alpha =-1/2$.

**Figure 22.**Illustration of the conjecture: plots of the shapes and contours of the RL copula density for $\alpha =-1/2$.

Name | Formula | Value |
---|---|---|

Gini gamma | $4{\int}_{0}^{1}\left[{C}_{\alpha}(x,x)+{C}_{\alpha}(x,1-x)\right]dx-2$ | $\frac{2({\alpha}^{2}+6\alpha +6)}{{\alpha}^{3}}-\frac{8(1+\alpha )({\alpha}^{2}+3\alpha +3)log(1+\alpha )}{{\alpha}^{4}(\alpha +2)}$ |

Spearman footrule coefficient | $6{\int}_{0}^{1}{C}_{\alpha}(x,x)dx-2$ | $\frac{2({\alpha}^{2}+12\alpha +12)}{{\alpha}^{3}}-\frac{12(1+\alpha )(\alpha +2)log(1+\alpha )}{{\alpha}^{4}}$ |

$\alpha $ | −0.5 | −0.3 | −0.1 | 0.0 | 0.6 | 1.2 | 1.8 | 2.4 | 3.0 |

$\rho $ | 0.3848 | 0.1549 | 0.038 | 0 | −0.1095 | −0.1429 | −0.1536 | −0.1556 | −0.1539 |

$\alpha $ | 0.01 | 0.31 | 0.61 | 0.91 | 1.21 | 1.51 |

$\rho $ | 0 | 0.0121 | 0.0482 | 0.1125 | 0.2133 | 0.3665 |

$\alpha $ | 0.01 | 0.51 | 1.01 | 1.51 | 2.01 | 2.51 | 3.01 | 3.51 | 4.01 | 4.51 |

$\rho $ | 0 | 0.0551 | 0.1495 | 0.2197 | 0.2594 | 0.2781 | 0.284 | 0.2827 | 0.2772 | 0.2695 |

$\alpha $ | 0.01 | 0.51 | 1.01 | 1.51 | 2.01 | 2.51 | 3.01 | 3.51 | 4.01 | 4.51 |

$\rho $ | 0.0017 | 0.0582 | 0.0867 | 0.1028 | 0.1123 | 0.118 | 0.1215 | 0.1235 | 0.1246 | 0.1249 |

**Table 6.**Values of the rho of Spearman of the RL copula for a grid of negative values for $\alpha $.

$\alpha $ | −0.50 | −0.35 | −0.20 | −0.05 |

$\rho $ | −0.1451 | −0.0833 | −0.0403 | −0.0087 |

Name | Abbreviation | Copula $((\mathit{x},\mathit{y},{\mathit{x}}_{\mathit{i}})\in {[0,1]}^{3})$ | $\mathit{\alpha}$ Domain | $\mathit{\rho}$ Domain |
---|---|---|---|---|

Ratio-polynomial | RP | $xy\frac{(1+\alpha )(1+\alpha xy)}{(1+\alpha x)(1+\alpha y)}$ | $\left[-\frac{1}{4},\infty \right)$ | $[-0.11,0.19]$ |

Second ratio-polynomial | RP | $xy\frac{(1+\alpha x)(1+\alpha y)}{(1+\alpha )(1+\alpha xy)}$ | $\left[-\frac{1}{2},\infty \right)$ | $[-0.16,0.39]$ |

Ratio-sine | RS | $xy\frac{sin\left(\alpha x\right)sin\left(\alpha y\right)}{sin\left(\alpha \right)sin\left(\alpha xy\right)}$ | $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ | $[0,0.37]$ |

Ratio-arctangent | RA | $xy\frac{arctan\left(\alpha x\right)arctan\left(\alpha y\right)}{arctan\left(\alpha \right)arctan\left(\alpha xy\right)}$ | $\mathbb{R}$ | $[0,0.27]$ |

Ratio-logarithmic | RL | $xy\frac{log(1+\alpha x)log(1+\alpha y)}{log(1+\alpha )log(1+\alpha xy)}$ | $\underset{\mathrm{Conjecture}:\phantom{\rule{4pt}{0ex}}[-1/2,\infty )}{[0,\infty )}$ | $\underset{\mathrm{Conjecture}:\phantom{\rule{4pt}{0ex}}[-0.15,0.13]}{[0,0.13]}$ |

Generalized RP | GRP | $\left(\prod _{i=1}^{n}{x}_{i}\right)\frac{{(1+\alpha )}^{n-1}\left(1+\alpha {\prod}_{i=1}^{n}{x}_{i}\right)}{{\prod}_{i=1}^{n}(1+\alpha {x}_{i})}$ | $\left[-\frac{1}{{2}^{n}},\infty \right)$ | − |

Inequality $((\mathit{x},\mathit{y},{\mathit{x}}_{\mathit{i}})\in {[0,1]}^{3})$ | $\mathit{\alpha}$ Domain |
---|---|

$(1+\alpha )xy(1+\alpha xy)\le min(x,y)(1+\alpha x)(1+\alpha y)$ | $\left[-\frac{1}{4},\infty \right)$ |

$max(x+y-1,0)(1+\alpha x)(1+\alpha y)\le (1+\alpha )xy(1+\alpha xy)$ | $\left[-\frac{1}{4},\infty \right)$ |

$xy(1+\alpha x)(1+\alpha y)\le min(x,y)(1+\alpha )(1+\alpha xy)$ | $\left[-\frac{1}{2},\infty \right)$ |

$max(x+y-1,0)(1+\alpha )(1+\alpha xy)\le xy(1+\alpha x)(1+\alpha y)$ | $\left[-\frac{1}{2},\infty \right)$ |

$xysin\left(\alpha x\right)sin\left(\alpha y\right)\le min(x,y)sin\left(\alpha \right)sin\left(\alpha xy\right)$ | $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ |

$max(x+y-1,0)sin\left(\alpha \right)sin\left(\alpha xy\right)\le xysin\left(\alpha x\right)sin\left(\alpha y\right)$ | $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ |

$xyarctan\left(\alpha x\right)arctan\left(\alpha y\right)\le min(x,y)arctan\left(\alpha \right)arctan\left(\alpha xy\right)$ | $\mathbb{R}$ |

$max(x+y-1,0)arctan\left(\alpha \right)arctan\left(\alpha xy\right)\le xyarctan\left(\alpha x\right)arctan\left(\alpha y\right)$ | $\mathbb{R}$ |

$xylog(1+\alpha x)log(1+\alpha y)\le min(x,y)log(1+\alpha )log(1+\alpha xy)$ | $[0,\infty )$ |

$max(x+y-1,0)log(1+\alpha )log(1+\alpha xy)\le xylog(1+\alpha x)log(1+\alpha y)$ | $[0,\infty )$ |

${(1+\alpha )}^{n-1}\left(\prod _{i=1}^{n}{x}_{i}\right)\left(1+\alpha \prod _{i=1}^{n}{x}_{i}\right)\le min({x}_{1},\dots ,{x}_{n})\prod _{i=1}^{n}(1+\alpha {x}_{i})$ | $\left[-\frac{1}{{2}^{n}},\infty \right)$ |

$max\left(1-n+\sum _{i=1}^{n}{x}_{i},0\right)\prod _{i=1}^{n}(1+\alpha {x}_{i})\le {(1+\alpha )}^{n-1}\left(\prod _{i=1}^{n}{x}_{i}\right)\left(1+\alpha \prod _{i=1}^{n}{x}_{i}\right)$ | $\left[-\frac{1}{{2}^{n}},\infty \right)$ |

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**MDPI and ACS Style**

Chesneau, C. Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form. *Modelling* **2023**, *4*, 102-132.
https://doi.org/10.3390/modelling4020008

**AMA Style**

Chesneau C. Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form. *Modelling*. 2023; 4(2):102-132.
https://doi.org/10.3390/modelling4020008

**Chicago/Turabian Style**

Chesneau, Christophe. 2023. "Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form" *Modelling* 4, no. 2: 102-132.
https://doi.org/10.3390/modelling4020008