# Bivariate Proportional Hazard Models: Structure and Inference

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## Abstract

**:**

## 1. Introduction

## 2. Bivariate Distributions with Proportional Hazard Marginals

- (i)
- Gumbel Type I distribution, with$$P({Y}_{1}>{y}_{1},{Y}_{2}>{y}_{2})=exp[-{\alpha}_{1}{y}_{1}-{\alpha}_{2}{y}_{2}-\delta {\alpha}_{1}{\alpha}_{2}{y}_{1}{y}_{2}],\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\le \delta \le 1.$$
- (ii)
- Gumbel Type II distribution, with$$P({Y}_{1}\le {y}_{1},{Y}_{2}\le {y}_{2})=[1-{e}^{-{\alpha}_{1}{y}_{1}}][1-{e}^{-{\alpha}_{2}{y}_{2}}][1+\delta {e}^{-{\alpha}_{1}{y}_{1}-{\alpha}_{2}{y}_{2}}],\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\delta \in [-1,1].$$
- (iii)
- Marshall–Olkin distribution (see Marshall and Olkin [1]), with$$P({Y}_{1}>{y}_{1},{Y}_{2}>{y}_{2})=exp[-{\alpha}_{1}{y}_{1}-{\alpha}_{2}{y}_{2}-\delta {\alpha}_{1}{\alpha}_{2}max({y}_{1},{y}_{2})],\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\delta \in [0,\infty ).$$

## 3. Bivariate Distributions with Proportional Hazard Conditionals

#### 3.1. The First Kind

#### 3.2. The Second Kind

## 4. If ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ Are Known

## 5. If ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ Are Known to Belong to Some Given Parametric Families

## 6. If ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ Are Unknown

## 7. If ${\mathit{F}}_{\mathbf{1}}$ and ${\mathit{F}}_{\mathbf{2}}$ Are Unknown in the PHC(I) Model

## 8. Application

- The Arnold and Strauss [3] bivariate exponential conditionals distribution. denoted by BEC.
- Gumbel’s [6] first bivariate exponential distribution, denoted by BG(I).
- The proportional hazard conditionals Weibull extension of the BEC distribution, denoted by PHC(I)-W.
- The proportional hazard conditionals Weibull extension of the BG(I) distribution, denoted by PHC(II)-W.

## 9. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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$\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\tau}}$ | $\widehat{\mathit{\delta}}$ | ${\widehat{\mathsf{\alpha}}}_{1}$ | ${\widehat{\mathsf{\alpha}}}_{2}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | n | RB | $\sqrt{\mathit{M}\mathit{S}\mathit{E}}$ | RB | $\sqrt{\mathit{M}\mathit{S}\mathit{E}}$ | RB | $\sqrt{\mathit{M}\mathit{S}\mathit{E}}$ | RB | $\sqrt{\mathit{M}\mathit{S}\mathit{E}}$ | RB | $\sqrt{\mathit{M}\mathit{S}\mathit{E}}$ |

30 | 0.1964 | 0.2630 | 0.4915 | 0.4029 | 1.6753 | 0.3424 | 0.2011 | 0.2591 | 0.5361 | 0.5931 | |

50 | 0.1911 | 0.2583 | 0.4864 | 0.3853 | 1.4262 | 0.2828 | 0.1983 | 0.2342 | 0.5339 | 0.5705 | |

(1.25, 0.75, 0.15, 1, 1) | 70 | 0.1842 | 0.2590 | 0.4819 | 0.3782 | 1.3480 | 0.2549 | 0.1974 | 0.2264 | 0.5330 | 0.5633 |

90 | 0.1717 | 0.2597 | 0.4751 | 0.3686 | 1.3117 | 0.2376 | 0.1933 | 0.2165 | 0.5307 | 0.5541 | |

30 | 0.1948 | 0.2675 | 0.4966 | 0.4078 | 1.6798 | 0.3393 | 0.1957 | 0.2542 | 0.5358 | 0.596 | |

50 | 0.1934 | 0.2620 | 0.4830 | 0.3823 | 1.4519 | 0.2841 | 0.1957 | 0.2310 | 0.5356 | 0.5731 | |

(1.25, 0.75, 0.30, 1, 1) | 70 | 0.1887 | 0.2611 | 0.4827 | 0.3771 | 1.3260 | 0.2514 | 0.1935 | 0.2238 | 0.5332 | 0.5653 |

90 | 0.1756 | 0.2579 | 0.4762 | 0.3688 | 1.2861 | 0.2351 | 0.1916 | 0.2178 | 0.5424 | 0.5629 | |

30 | 0.1894 | 0.2610 | 0.5316 | 0.4336 | 0.4177 | 0.3002 | 0.1918 | 0.2463 | 0.6127 | 0.6808 | |

50 | 0.1827 | 0.2512 | 0.5149 | 0.4090 | 0.4022 | 0.2688 | 0.1904 | 0.2297 | 0.6018 | 0.6428 | |

(1.25, 0.75, 0.45, 1, 1) | 70 | 0.1799 | 0.2483 | 0.4966 | 0.3884 | 0.4009 | 0.2604 | 0.1900 | 0.2210 | 0.5904 | 0.6193 |

90 | 0.1679 | 0.2512 | 0.4905 | 0.3799 | 0.3958 | 0.2376 | 0.1779 | 0.2126 | 0.5878 | 0.6136 | |

30 | 0.4228 | 0.7457 | 1.2543 | 0.6503 | 1.6778 | 0.3378 | 0.4287 | 0.4473 | 1.4193 | 1.4930 | |

50 | 0.4189 | 0.7427 | 1.2332 | 0.6305 | 1.5481 | 0.2941 | 0.4270 | 0.4353 | 1.4191 | 14.435 | |

(1.75, 0.5, 0.15, 1, 1) | 70 | 0.4193 | 0.7422 | 1.2311 | 0.6262 | 1.3103 | 0.2455 | 0.4245 | 0.4347 | 1.4048 | 1.4378 |

90 | 0.4093 | 0.7326 | 1.2276 | 0.6215 | 1.2531 | 0.2294 | 0.4227 | 0.4346 | 1.3939 | 1.4363 | |

30 | 0.4218 | 0.7455 | 1.2862 | 0.6652 | 0.8103 | 0.3439 | 0.4204 | 0.4352 | 1.4968 | 1.5718 | |

50 | 0.4207 | 0.7428 | 1.2554 | 0.6408 | 0.7520 | 0.2984 | 0.4152 | 0.4288 | 1.4907 | 1.5178 | |

(1.75, 0.5, 0.30, 1, 1) | 70 | 0.4171 | 0.7395 | 1.2544 | 0.6373 | 0.6981 | 0.2686 | 0.4154 | 0.4281 | 1.4804 | 1.5167 |

90 | 0.4098 | 0.7334 | 1.2421 | 0.6300 | 0.6730 | 0.2539 | 0.4143 | 0.4250 | 1.4733 | 1.5102 | |

30 | 0.4179 | 0.7376 | 1.2780 | 0.6602 | 0.4190 | 0.2985 | 0.4149 | 0.4270 | 1.5069 | 1.5796 | |

50 | 0.4169 | 0.7361 | 1.2606 | 0.6456 | 0.4115 | 0.2725 | 0.4148 | 0.4250 | 1.4762 | 1.5240 | |

(1.75, 0.5, 0.45, 1, 1) | 70 | 0.4126 | 0.7326 | 1.2354 | 0.6268 | 0.4056 | 0.2597 | 0.4122 | 0.4239 | 1.4655 | 1.4988 |

90 | 0.4034 | 0.7225 | 1.2303 | 0.6238 | 0.3916 | 0.2455 | 0.4062 | 0.4227 | 1.4550 | 1.4831 | |

30 | 0.3903 | 0.3270 | 0.2506 | 0.3899 | 1.3177 | 0.3091 | 0.3590 | 0.4395 | 0.2778 | 0.3212 | |

50 | 0.3661 | 0.2980 | 0.2434 | 0.3887 | 1.1349 | 0.2508 | 0.3479 | 0.4006 | 0.2776 | 0.3044 | |

(0.75, 1.5, 0.15, 1, 1) | 70 | 0.3516 | 0.2811 | 0.2493 | 0.3877 | 1.1461 | 0.2343 | 0.3471 | 0.3816 | 0.2761 | 0.2962 |

90 | 0.3452 | 0.2721 | 0.2305 | 0.3871 | 1.0392 | 0.2106 | 0.3449 | 0.3755 | 0.2748 | 0.2929 | |

30 | 0.3877 | 0.3280 | 0.2439 | 0.3831 | 0.6461 | 0.3136 | 0.3794 | 0.4639 | 0.2677 | 0.3001 | |

50 | 0.3699 | 0.3006 | 0.2424 | 0.3817 | 0.6403 | 0.2784 | 0.3539 | 0.4050 | 0.2660 | 0.2871 | |

(0.75, 1.5, 0.30, 1, 1) | 70 | 0.3568 | 0.2861 | 0.2365 | 0.3785 | 0.6328 | 0.2611 | 0.3527 | 0.3896 | 0.2600 | 0.2884 |

90 | 0.3532 | 0.2784 | 0.2273 | 0.3767 | 0.6066 | 0.2373 | 0.3513 | 0.3819 | 0.2492 | 0.2823 | |

30 | 0.3933 | 0.3324 | 0.2529 | 0.3914 | 0.3934 | 0.2906 | 0.3774 | 0.4611 | 0.2664 | 0.3023 | |

50 | 0.3685 | 0.2996 | 0.2439 | 0.3900 | 0.3893 | 0.2662 | 0.3653 | 0.4186 | 0.2639 | 0.2872 | |

(0.75, 1.5, 0.45, 1, 1) | 70 | 0.3557 | 0.2834 | 0.2495 | 0.3883 | 0.3649 | 0.2523 | 0.3595 | 0.3955 | 0.2594 | 0.2856 |

90 | 0.3482 | 0.2747 | 0.2286 | 0.3861 | 0.3486 | 0.2379 | 0.3551 | 0.3833 | 0.2526 | 0.2829 |

Estimates | BG(I) | PHC(II)-W | BEC | PHC(I)-W |
---|---|---|---|---|

$\widehat{{\alpha}_{1}}$ | 0.2198 | 0.0775 | 0.0940 | 0.0277 |

(0.0185) | (0.0389) | (0.0081) | (0.0077) | |

$\widehat{{\alpha}_{2}}$ | 0.0656 | 0.0055 | 0.0283 | 0.0046 |

(0.0056) | (0.0029) | (0.0089) | (0.0008) | |

$\widehat{\delta}$ | 0.8344 | 0.9053 | 3.8624 | 0.8753 |

(0.3041) | (0.3341) | (0.3254) | (0.0006) | |

$\widehat{\theta}$ | 1.6396 | 2.0337 | ||

(0.2979) | (0.0007) | |||

$\widehat{\tau}$ | 1.8895 | 1.8384 | ||

(0.1995) | (0.0006) | |||

AIC | 400.6863 | 364.4660 | 415.2929 | 353.4773 |

BIC | 405.1758 | 371.9485 | 419.7824 | 360.9598 |

PHC(II)-W vs. BG(I) | PHC(I)-W vs. BEC | |
---|---|---|

$-2log(\Lambda )$ | 40.2203 | 65.8156 |

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**MDPI and ACS Style**

Arnold, B.C.; Martínez-Flórez, G.; Gómez, H.W. Bivariate Proportional Hazard Models: Structure and Inference. *Symmetry* **2022**, *14*, 2073.
https://doi.org/10.3390/sym14102073

**AMA Style**

Arnold BC, Martínez-Flórez G, Gómez HW. Bivariate Proportional Hazard Models: Structure and Inference. *Symmetry*. 2022; 14(10):2073.
https://doi.org/10.3390/sym14102073

**Chicago/Turabian Style**

Arnold, Barry C., Guillermo Martínez-Flórez, and Héctor W. Gómez. 2022. "Bivariate Proportional Hazard Models: Structure and Inference" *Symmetry* 14, no. 10: 2073.
https://doi.org/10.3390/sym14102073