# The Copula Derived from the SAHARA Utility Function

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**1.**

**Proposition**

**2.**

## 3. SAHARA Family

**Remark**

**1.**

## 4. Application

- 4.2.20 Nelsen:$\phantom{\rule{1.em}{0ex}}{\psi}_{\theta}\left(t\right)={\left\{log\left[e+t\right]\right\}}^{-\frac{1}{\theta}},\phantom{\rule{1.em}{0ex}}\theta >0$.
- Special:$\phantom{\rule{1.em}{0ex}}{\psi}_{\theta}\left(t\right)={\left(\frac{-t+\sqrt{{t}^{2}+4}}{2}\right)}^{\frac{1}{\theta}},\phantom{\rule{1.em}{0ex}}\theta >0$.

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Graphical comparison between theoretical $\lambda \left(w\right)=w-K\left(w\right)$ for SAHARA (yellow), BB2 (green) and 4.2.20 Nelsen (orange) and empirical one (blue).

Copula | Parameter Estimates | Error ${\mathsf{\phi}}_{\widehat{\mathsf{\theta}}}^{\mathbf{[}\mathbf{-}\mathbf{1}\mathbf{]}}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}$ |
---|---|---|

4.2.20 Nelsen | $\widehat{\theta}=1.005$ | $0.720$ |

BB2 Joe (1997) | $\widehat{\theta}=1.469$; $\widehat{\u03f5}=0.383$ | $0.667$ |

SAHARA | $\widehat{\theta}=0.204$; $\widehat{\u03f5}=0.914$ | $0.293$ |

Deaths | Exposure | Mortality | |
---|---|---|---|

Partner dead | |||

$e=0$ | 69 | 604.87 | 0.114075 |

$e=1$ | 17 | $428.44$ | $0.039679$ |

$e=2$ | 9 | $277.76$ | $0.032403$ |

$e=3$ | 4 | $155.08$ | $0.025590$ |

$e=4$ | 3 | $49.67$ | $0.060395$ |

Partner alive | 751 | 34,631.45 | $0.021685$ |

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Spreeuw, J.
The Copula Derived from the SAHARA Utility Function. *Risks* **2022**, *10*, 133.
https://doi.org/10.3390/risks10070133

**AMA Style**

Spreeuw J.
The Copula Derived from the SAHARA Utility Function. *Risks*. 2022; 10(7):133.
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2022. "The Copula Derived from the SAHARA Utility Function" *Risks* 10, no. 7: 133.
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