A Generalization of Binomial Exponential-2 Distribution: Copula, Properties and Applications
Abstract
:1. Introduction
2. The New Model and Its Motivation
3. Copula under the EBE Model
3.1. Bivariate EBE (BivEBE) Type via Renyi’s Entropy
3.2. BivEBE Type Using “Farlie-Gumbel-Morgenstern” (FGM) Copula
3.3. BivEBE Type via “Modified FGM” (MFGM) Copula
3.3.1. BivEBE-FGM (Type-I) Model
3.3.2. BivEBE-FGM (Type-II) Model
3.3.3. BivEBE-FGM (Type-III) Model
3.3.4. BivEBE-FGM (Type-IV) Model
3.4. BivEBE Type via Clayton Copula
4. Properties
4.1. Expansions and Quantile Function (QF)
4.2. Moments
4.3. Incomplete Moments
4.4. Mean Deviation and Bonferroni and Lorenz Curve
4.5. Residual Life and Reversed Residual Life Functions
5. Estimation and Inference
6. Simulation
7. Modeling Stress-Rupture Life of Kevlar 49/Epoxy Strands Data
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Sample Availability: The data used to support the findings in this study are included within the paper. |
α | θ | λ | μ1′ | μ2′ | μ3′ | μ4′ | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|
0.5 | 0.5 | 0.5 | 1.066 | 3.009 | 16.07 | 149.755 | 2.752 | 12.349 |
0.5 | 0.7 | 0.7 | 1.660 | 4.313 | 23.29 | 277.509 | 2.189 | 9.1470 |
0.6 | 0.7 | 1.5 | 2.414 | 4.008 | 14.47 | 125.278 | 1.669 | 6.8000 |
0.7 | 0.3 | 2.7 | 2.273 | 2.749 | 8.148 | 60.4810 | 1.627 | 6.6450 |
1.5 | 0.5 | 0.5 | 0.243 | 0.316 | 0.545 | 1.58900 | 2.753 | 12.231 |
1.5 | 0.2 | 0.7 | 0.381 | 0.294 | 0.458 | 1.26900 | 2.539 | 11.139 |
2.6 | 0.2 | 1.5 | 0.404 | 0.149 | 0.121 | 0.23300 | 1.929 | 8.0020 |
2.7 | 0.4 | 2.7 | 0.626 | 0.195 | 0.153 | 0.32500 | 1.571 | 6.3250 |
n | λ | α | θ | MSEs | |
---|---|---|---|---|---|
50 | 0.5 | 0.5 | 0.5 | 0.11892, 0.23647, 0.06169 | |
100 | 0.08304, 0.17158, 0.04192 | ||||
200 | 0.05794, 0.12764, 0.02949 | ||||
500 | 0.03636, 0.08476, 0.01866 | ||||
1000 | 0.02595, 0.06105, 0.01325 | ||||
50 | 0.5 | 0.7 | 0.7 | 0.08813, 0.17030, 0.08578 | |
100 | 0.06136, 0.11879, 0.05894 | ||||
200 | 0.04320, 0.08131, 0.04139 | ||||
500 | 0.02764, 0.04874, 0.02616 | ||||
1000 | 0.01964, 0.03346, 0.01856 | ||||
50 | 0.6 | 0.7 | 1.5 | 0.06866, 0.12976, 0.18044 | |
100 | 0.04885, 0.08743, 0.12657 | ||||
200 | 0.03427, 0.06270, 0.08909 | ||||
500 | 0.02180, 0.03709, 0.05538 | ||||
1000 | 0.01550, 0.02515, 0.03903 | ||||
50 | 0.7 | 0.3 | 2.7 | 0.07211, 0.13178, 0.29484 | |
100 | 0.05105, 0.09550, 0.20625 | ||||
200 | 0.03537, 0.07094, 0.14854 | ||||
500 | 0.02227, 0.04645, 0.09524 | ||||
1000 | 0.01545, 0.03457, 0.06935 | ||||
50 | 0.5 | 0.5 | 0.5 | 0.11892, 0.23647, 0.06169 | |
100 | 0.08304, 0.17158, 0.04192 | ||||
200 | 0.05794, 0.12764, 0.02949 | ||||
500 | 0.03636, 0.08476, 0.01866 | ||||
1000 | 0.02595, 0.06105, 0.01325 |
n | λ | α | θ | MSEs | |
---|---|---|---|---|---|
50 | 0.5 | 0.5 | 0.5 | 0.18356, 0.31326, 0.06034 | |
100 | 0.14037, 0.29166, 0.02491 | ||||
200 | 0.10718, 0.27047, 0.00952 | ||||
500 | 0.07427, 0.22547, 0.00392 | ||||
1000 | 0.05854, 0.18297, 0.00431 | ||||
50 | 0.5 | 0.7 | 0.7 | 0.13103, 0.32039, 0.16804 | |
100 | 0.09571, 0.29243, 0.12804 | ||||
200 | 0.07617, 0.25160, 0.11495 | ||||
500 | 0.05278, 0.17909, 0.10177 | ||||
1000 | 0.03559, 0.11151, 0.09647 | ||||
50 | 0.6 | 0.7 | 1.5 | 0.11728, 0.32816, 1.12796 | |
100 | 0.09252, 0.29394, 1.03834 | ||||
200 | 0.07715, 0.27919, 0.98811 | ||||
500 | 0.05480, 0.21775, 0.91763 | ||||
1000 | 0.03922, 0.17326, 0.88793 | ||||
50 | 0.7 | 0.3 | 2.7 | 0.16086, 0.36788, 2.90744 | |
100 | 0.13715, 0.35482, 2.70484 | ||||
200 | 0.10661,0.32348, 2.66995 | ||||
500 | 0.08491, 0.27404, 2.64048 | ||||
1000 | 0.07009, 0.22686, 2.72810 | ||||
50 | 0.5 | 0.5 | 0.5 | 0.18356, 0.31326, 0.06034 | |
100 | 0.14037, 0.29166, 0.02491 | ||||
200 | 0.10718, 0.27047, 0.00952 | ||||
500 | 0.07427, 0.22547, 0.00392 | ||||
1000 | 0.05854, 0.18297, 0.00431 |
n | λ | α | θ | MSEs | |
---|---|---|---|---|---|
50 | 0.5 | 0.5 | 0.5 | 0.55465, 0.50616, 0.52281 | |
100 | 0.53818, 0.51109, 0.50207 | ||||
200 | 0.52237, 0.48520, 0.49930 | ||||
500 | 0.50325, 0.47041, 0.49911 | ||||
1000 | 0.49813, 0.46471, 0.50127 | ||||
50 | 0.5 | 0.7 | 0.7 | 0.50550, 0.58874, 0.72531 | |
100 | 0.49190, 0.60765, 0.70470 | ||||
200 | 0.48878, 0.63157, 0.69991 | ||||
500 | 0.49378, 0.66331, 0.69945 | ||||
1000 | 0.49624, 0.67996, 0.70161 | ||||
50 | 0.6 | 0.7 | 1.5 | 0.57701, 0.58491, 1.52331 | |
100 | 0.58530, 0.61671, 1.51090 | ||||
200 | 0.57826, 0.61435, 1.50499 | ||||
500 | 0.58310, 0.65758, 1.47994 | ||||
1000 | 0.58988, 0.68285, 1.47548 | ||||
50 | 0.7 | 0.3 | 2.7 | 0.74776, 0.45747, 2.49293 | |
100 | 0.74369, 0.44427, 2.46686 | ||||
200 | 0.72548, 0.39578, 2.51294 | ||||
500 | 0.71662, 0.36901, 2.54710 | ||||
1000 | 0.70309, 0.31662, 2.62308 | ||||
50 | 0.5 | 0.5 | 0.5 | 0.55465, 0.50616, 0.52281 | |
100 | 0.53818, 0.51109, 0.50207 | ||||
200 | 0.52237, 0.48520, 0.49930 | ||||
500 | 0.50325, 0.47041, 0.49911 | ||||
1000 | 0.49813, 0.46471, 0.50127 |
n | λ | α | θ | MSEs | |
---|---|---|---|---|---|
50 | 0.5 | 0.5 | 0.5 | 0.46616, 0.92696, 0.24180 | |
100 | 0.32550, 0.67258, 0.16431 | ||||
200 | 0.22713, 0.50034, 0.11560 | ||||
500 | 0.14251, 0.33226, 0.07314 | ||||
1000 | 0.10172, 0.23930, 0.05195 | ||||
50 | 0.5 | 0.7 | 0.7 | 0.34545, 0.66757, 0.33625 | |
100 | 0.24054, 0.46565, 0.23104 | ||||
200 | 0.16935, 0.31873, 0.16226 | ||||
500 | 0.10834, 0.19105, 0.10255 | ||||
1000 | 0.07700, 0.13116, 0.07274 | ||||
50 | 0.6 | 0.7 | 1.5 | 0.26914, 0.50864, 0.70733 | |
100 | 0.19150, 0.34271, 0.49614 | ||||
200 | 0.13434, 0.24579, 0.34924 | ||||
500 | 0.08545, 0.14539, 0.21710 | ||||
1000 | 0.06077, 0.09860, 0.15300 | ||||
50 | 0.7 | 0.3 | 2.7 | 0.28268, 0.51657, 1.15574 | |
100 | 0.20011, 0.37434, 0.80847 | ||||
200 | 0.13865, 0.27809, 0.58226 | ||||
500 | 0.08730, 0.18209, 0.37332 | ||||
1000 | 0.06055, 0.13552, 0.27184 | ||||
50 | 0.5 | 0.5 | 0.5 | 0.46616, 0.92696, 0.24180 | |
100 | 0.32550, 0.67258, 0.16431 | ||||
200 | 0.22713, 0.50034, 0.11560 | ||||
500 | 0.14251, 0.33226, 0.07314 | ||||
1000 | 0.10172, 0.23930, 0.05195 |
n | λ | α | θ | MSEs | |
---|---|---|---|---|---|
50 | 0.5 | 0.5 | 0.5 | 0.82686, 0.71564, 0.79176 | |
100 | 0.76642, 0.65414, 0.75986 | ||||
200 | 0.69834, 0.57143, 0.79418 | ||||
500 | 0.66517, 0.51762, 0.82334 | ||||
1000 | 0.57302, 0.51648, 0.81022 | ||||
50 | 0.5 | 0.7 | 0.7 | 0.81610, 0.73529, 0.76823 | |
100 | 0.78664, 0.58513, 0.72796 | ||||
200 | 0.73695, 0.52505, 0.62213 | ||||
500 | 0.72222, 0.47172, 0.60909 | ||||
1000 | 0.74675, 0.49449, 0.61361 | ||||
50 | 0.6 | 0.7 | 1.5 | 0.73690, 0.54738, 0.00000 | |
100 | 0.70523, 0.40712, 0.00000 | ||||
200 | 0.65622, 0.34796, 0.00000 | ||||
500 | 00.6434, 0.28687, 0.00000 | ||||
1000 | 0.65265, 0.25726, 0.00000 | ||||
50 | 0.7 | 0.3 | 2.7 | 0.63454, 0.50188, 0.00000 | |
100 | 0.51110, 0.28606, 0.00000 | ||||
200 | 0.42228, 0.23378, 0.00000 | ||||
500 | 0.31797, 0.22104, 0.00000 | ||||
1000 | 0.29115, 0.20381, 0.00000 | ||||
50 | 0.5 | 0.5 | 0.5 | 0.82686, 0.71564, 0.79176 | |
100 | 0.76642, 0.65414, 0.75986 | ||||
200 | 0.69834, 0.57143, 0.79418 | ||||
500 | 0.66517, 0.51762, 0.82334 | ||||
1000 | 0.57302, 0.51648, 0.81022 |
Model | Estimates | ||||
---|---|---|---|---|---|
Burr X | a | ||||
0.462891 | |||||
Burr XII | a | b | |||
1.14571 | 1.94994 | ||||
ELLG | a | b | |||
1.211659 | 0.570392 | ||||
BLLG | a | b | c | ||
0.581650 | 1.091929 | 1.295956 | |||
EBE | α | θ | λ | ||
0.803609 | 0.001 | 0.999053 | |||
BLLGW | a | b | c | α | β |
6.468097 | 5.07658 | 0.22372 | 0.24463 | 0.93981 | |
BWLLG | a | b | c | α | β |
0.70773 | 0.15311 | 1.45779 | 5.74192 | 0.72356 |
Model | −2 logL | ||||
---|---|---|---|---|---|
EBE | 143.3996 | 149.4100 | 152.3784 | 152.3784 | 149.6087 |
Burr XII | 145.4801 | 149.4801 | 154.4119 | 152.4660 | 149.6229 |
BLLGW | 204.0771 | 214.0771 | 227.1527 | 219.3705 | 214.7087 |
BWLLG | 204.8205 | 214.8205 | 227.8961 | 220.1139 | 215.4521 |
Burr X | 285.8730 | 287.8730 | 290.3389 | 288.8659 | 287.9200 |
ELLG | 587.6830 | 591.6830 | 596.9133 | 593.8004 | 591.8055 |
BLLG | 462.1078 | 468.1078 | 175.9531 | 471.2838 | 468.3552 |
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Alotaibi, N.; Malyk, I.V. A Generalization of Binomial Exponential-2 Distribution: Copula, Properties and Applications. Symmetry 2020, 12, 1338. https://doi.org/10.3390/sym12081338
Alotaibi N, Malyk IV. A Generalization of Binomial Exponential-2 Distribution: Copula, Properties and Applications. Symmetry. 2020; 12(8):1338. https://doi.org/10.3390/sym12081338
Chicago/Turabian StyleAlotaibi, Naif, and Igor V. Malyk. 2020. "A Generalization of Binomial Exponential-2 Distribution: Copula, Properties and Applications" Symmetry 12, no. 8: 1338. https://doi.org/10.3390/sym12081338