A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas
Abstract
:1. Introduction
2. The Half-Logistic Copula
- i.
- The product copula is a limit case of the half-logistic copula when θ approaches zero.
- ii.
- The Fréchet–Hoeffding upper bound copula is a limit case of the half-logistic copula when θ tends to infinity.
3. Dependence
- i.
- Stochastically increasing, thus left-tail decreasing and right-tail increasing, hence positive quadrant-dependent;
- ii.
- is , and the survival copula is also ;
- iii.
- , where denotes Spearman’s rho correlation coefficient;
- iv.
- .
4. The Two-Parameter Half-logistic Copula
- is strictly increasing, because its first derivative is positive for all .
- is log-concave. As
- and .
5. Applications
5.1. St-Maurice’s River Annual Flow Data
5.2. Shunters’ Accidents Data
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proofs
Appendix A.1. Proof of Theorem 3
- i.
- To prove this, we use (Nelsen [16], Theorem 4.4.7, p. 139). The multiplicative Archimedean generator of the product copula is ; thus, the product copula will be the limit of at if We haveTherefore,
- ii.
- To prove this, assume . Therefore, we have
Appendix A.2. Proof of Theorem 4
Appendix A.3. Proof of Theorem 5
Appendix A.4. Proof of Theorem 6
Appendix A.5. Proof of Theorem 7
Appendix A.6. Proof of Theorem 8
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Copula | Kendall’s Estimator | MLE | ||
---|---|---|---|---|
(p-Value) | (p-Value) | (p-Value) | (p-Value) | |
Half-logistic | 0.0964 (0.21) | 0.8251 (0.03) | 0.0971 (0.19) | 0.8288 (0.52) |
Frank | 0.0790 (0.34) | 0.7253 (0.08) | 0.0788 (0.43) | 0.7238 (0.07) |
Clayton | 0.0879 (0.25) | 0.2494 (0.84) | 0.4488 (0.03) | 1.0006 (0.05) |
Gumbel–Hougaard | 0.0688 (0.45) | 0.7571 (0.05) | 0.0573 (0.71) | 0.7068 (0.05) |
Product | 0.9679 (0.00) | 1.4608 (0.00) |
Copula | AIC | BIC | ||||||
---|---|---|---|---|---|---|---|---|
Half-logistic | 767.2 | 775.4 | 5.113 | 0.227 | 5.949 | 0.121 | 6.036 | |
Survival half-logistic | 775.2 | 783.4 | 5.090 | 0.247 | 5.937 | 0.130 | 7.166 | |
Frank | 767 | 775.1 | 5.116 | 0.226 | 5.952 | 0.119 | 5.515 | |
Clayton | 773.9 | 782.1 | 5.105 | 0.235 | 5.952 | 0.119 | 1.073 | |
Gumbel–Hougaard | 769.6 | 777.8 | 5.101 | 0.216 | 5.951 | 0.125 | 0.421 | |
Two-parameter half-logistic | 769 | 778.8 | 5.116 | 0.226 | 5.952 | 0.120 | 5.581 | 0.101 |
Model | AIC | BIC | ||
---|---|---|---|---|
FGM | 686.1 | 688.9 | 0.957 | |
Frank | 686 | 688.8 | 2.030 | |
Truncated Poisson | 690 | 692.8 | 5.861 | |
Half-logistic | 685.2 | 688 | 2.703 | |
Survival half-logistic | 736.4 | 739.2 | 4.503 | |
Two-parameter half-logistic | 686.8 | 692.4 | 3.401 | 3.008 |
Model | Absolute Distance | Squared Distance |
---|---|---|
Frank | 0.538 | 0.006 |
Half-logistic | 0.486 | 0.005 |
Truncated Poisson | 0.551 | 0.007 |
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Alzaid, A.A.; Alhadlaq, W.M. A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas. Mathematics 2024, 12, 101. https://doi.org/10.3390/math12010101
Alzaid AA, Alhadlaq WM. A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas. Mathematics. 2024; 12(1):101. https://doi.org/10.3390/math12010101
Chicago/Turabian StyleAlzaid, Abdulhamid A., and Weaam M. Alhadlaq. 2024. "A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas" Mathematics 12, no. 1: 101. https://doi.org/10.3390/math12010101
APA StyleAlzaid, A. A., & Alhadlaq, W. M. (2024). A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas. Mathematics, 12(1), 101. https://doi.org/10.3390/math12010101