The Generalized Exponential Extended Exponentiated Family of Distributions: Theory, Properties, and Applications
Abstract
:1. Introduction
- 1.
- , is differentiable and increases monotonically.
- 2.
1.1. Reliability Analysis
1.2. Useful Expansion of the NGE3-G Family
2. Statistical Properties of NGE3-G Family (NGE3GF)
2.1. Quantile Function (QF)
2.2. Skewness and Kurtosis
2.3. Probability Weighted Moments (PWMs)
2.4. Moments
2.5. Moment Generating Function (mgf)
2.6. Mean Deviation (MD)
2.7. Order Statistics (OS)
2.8. Renyi Entropy (RE)
3. Maximum Likelihood (ML) Method
4. NGE3-Exponential Distribution (NGE3ED)
4.1. Relationship among Submodels of NGE3ED
- 1.
- For , the NGE3E () model reduces to new generalized extended exponentiated exponential NGE2E () model.
- 2.
- For , the NGE3E () model reduces to new generalized exponential extended exponential model NGEEE ().
- 3.
- For , the NGE3E () model reduces to new generalized extended exponential model NGEE ().
- 4.
- For , the NGE3E () model reduces to new generalized exponential model NGE-1 ().
- 5.
- For , the NGE3E () model reduces to the NGE-2 model ().
4.2. Quantile Function and Simulation Study
4.3. Estimation
4.4. Empirical Illustrations of the NGE3E Model
4.4.1. Dataset 1: Guinea Pig (GP) Data
0.10 | 0.33 | 0.44 | 0.56 | 0.59 | 0.72 | 0.74 | 0.77 | 0.92 | 0.93 | |
0.96 | 1.0 | 1.0 | 1.02 | 1.05 | 1.07 | 7.0 | 0.08 | 1.08 | 1.08 | 1.09 |
1.12 | 1.13 | 1.15 | 1.16 | 1.20 | 1.21 | 1.22 | 1.22 | 1.24 | 1.30 | 1.34 |
1.36 | 1.39 | 1.44 | 1.46 | 1.53 | 1.59 | 1.60 | 1.63 | 1.63 | 1.68 | 1.71 |
1.72 | 1.76 | 1.83 | 1.95 | 1.96 | 1.97 | 2.02 | 2.13 | 2.15 | 2.16 | 2.22 |
2.30 | 2.31 | 2.40 | 2.45 | 2.51 | 2.53 | 2.54 | 2.54 | 2.78 | 2.93 | 3.27 |
3.42 | 3.47 | 3.61 | 4.02 | 4.32 | 4.58 | 5.55 |
4.4.2. Dataset 2: Bladder Cancer (BC) Data
0.08 | 2.09 | 22.69 | 12.63 | 3.48 | 4.87 | 8.65 | 6.93 | 6.94 | 8.66 | |
3.36 | 2.07 | 13.11 | 23.63 | 21.73 | 12.07 | 0.20 | 2.23 | 6.76 | 3.36 | 3.52 |
4.98 | 2.02 | 20.28 | 6.97 | 9.02 | 12.03 | 8.53 | 13.29 | 0.40 | 6.54 | 4.51 |
2.26 | 3.57 | 3.31 | 2.02 | 5.06 | 7.09 | 9.22 | 12.02 | 13.80 | 8.37 | 25.74 |
6.25 | 0.50 | 4.50 | 2.46 | 3.25 | 3.46 | 1.76 | 5.09 | 19.13 | 7.26 | 11.98 |
9.47 | 8.26 | 14.24 | 5.85 | 25.82 | 4.40 | 0.51 | 1.46 | 2.54 | 18.10 | 3.70 |
11.79 | 5.17 | 7.93 | 7.28 | 5.71 | 9.74 | 4.34 | 14.76 | 3.02 | 26.31 | 1.40 |
0.81 | 4.33 | 17.36 | 2.83 | 11.64 | 1.26 | 7.87 | 46.12 | 5.62 | 17.12 | 2.87 |
7.63 | 1.35 | 79.05 | 5.41 | 17.14 | 4.26 | 11.25 | 2.75 | 7.66 | 1.19 | 5.49 |
43.01 | 10.66 | 16.62 | 7.59 | 5.34 | 4.18 | 10.75 | 7.62 | 2.69 | 5.41 | 0.90 |
4.23 | 34.26 | 2.69 | 14.83 | 1.05 | 10.34 | 36.66 | 7.39 | 15.96 | 5.32 | 3.88 |
2.64 | 32.15 | 14.77 | 10.06 | 7.32 | 5.32 | 3.82 | 2.62 |
- The EWE distribution
- WE distribution:
- EEW distribution:
- BEF distribution:
- OGEW distribution:
- OGEFr distribution:
- EKE distribution:
- KwW distribution:
- WL distribution:
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Set I: | Set II: | ||||||
---|---|---|---|---|---|---|---|
n | Parameter | MLE | MSE | AB | MLE | MSE | AB |
50 | 4.178052 | 0.713267 | 0.078052 | 3.566745 | 1.041872 | 0.266745 | |
2.639318 | 2.241914 | 0.350682 | 3.607051 | 4.074313 | 1.272949 | ||
2.996142 | 2.892954 | 0.886142 | 2.542916 | 4.662988 | 1.022916 | ||
0.222999 | 0.007290 | 0.012999 | 0.383463 | 0.042189 | 0.073464 | ||
300 | 4.322833 | 0.380854 | 0.222833 | 3.625264 | 0.460963 | 0.325264 | |
2.709852 | 1.547025 | 0.280148 | 3.434403 | 4.584252 | 1.445597 | ||
2.243113 | 0.918195 | 0.133113 | 1.703476 | 0.713168 | 0.183476 | ||
0.243876 | 0.006906 | 0.033876 | 0.432456 | 0.047154 | 0.122456 | ||
550 | 4.265227 | 0.273629 | 0.165227 | 3.526255 | 0.245441 | 0.226255 | |
2.852406 | 1.260532 | 0.137594 | 3.694286 | 3.433749 | 1.185714 | ||
2.194856 | 0.586033 | 0.084856 | 1.610212 | 0.311643 | 0.090212 | ||
0.233397 | 0.004786 | 0.023397 | 0.398196 | 0.026787 | 0.088196 | ||
800 | 4.238538 | 0.217878 | 0.138538 | 3.471447 | 0.156626 | 0.171447 | |
2.892669 | 1.054043 | 0.097331 | 3.875455 | 2.720822 | 1.004546 | ||
2.168020 | 0.431818 | 0.058020 | 1.581829 | 0.180927 | 0.061829 | ||
0.228880 | 0.003738 | 0.018880 | 0.378363 | 0.017620 | 0.068363 | ||
1050 | 4.211064 | 0.170239 | 0.111064 | 3.446462 | 0.113253 | 0.146462 | |
2.919116 | 0.895349 | 0.070884 | 3.971339 | 2.321943 | 0.908661 | ||
2.157062 | 0.334494 | 0.047062 | 1.558690 | 0.119622 | 0.038690 | ||
0.225160 | 0.002968 | 0.015160 | 0.368524 | 0.012958 | 0.058524 | ||
1300 | 4.197459 | 0.140223 | 0.097459 | 3.416337 | 0.084250 | 0.116337 | |
2.926014 | 0.764960 | 0.063986 | 4.078104 | 1.970276 | 0.801896 | ||
2.141350 | 0.257725 | 0.031350 | 1.564415 | 0.101034 | 0.044415 | ||
0.223226 | 0.002444 | 0.013226 | 0.358211 | 0.009942 | 0.048211 | ||
1550 | 4.172582 | 0.116356 | 0.072582 | 3.407995 | 0.071701 | 0.107995 | |
2.938682 | 0.651132 | 0.051318 | 4.132392 | 1.763364 | 0.747608 | ||
2.150551 | 0.227299 | 0.040551 | 1.553020 | 0.080565 | 0.033020 | ||
0.220265 | 0.002023 | 0.010265 | 0.354000 | 0.008316 | 0.044000 | ||
1800 | 4.174525 | 0.104986 | 0.074525 | 3.391980 | 0.056767 | 0.091980 | |
2.938528 | 0.594311 | 0.051472 | 4.200579 | 1.543466 | 0.679421 | ||
2.136263 | 0.198713 | 0.026263 | 1.547978 | 0.065708 | 0.027978 | ||
0.220122 | 0.001831 | 0.010122 | 0.348373 | 0.006657 | 0.038373 | ||
2050 | 4.164815 | 0.092521 | 0.064815 | 3.382171 | 0.049089 | 0.082171 | |
2.940650 | 0.523406 | 0.049350 | 4.242065 | 1.409728 | 0.637934 | ||
2.138791 | 0.175037 | 0.028791 | 1.550037 | 0.059096 | 0.030037 | ||
0.218764 | 0.001610 | 0.008764 | 0.344838 | 0.005818 | 0.034838 |
Dataset | Min. | Median | Max. | Mean | Variance | Skewness | Kurtosis | ||
---|---|---|---|---|---|---|---|---|---|
1 | 0.080 | 1.080 | 1.560 | 2.303 | 7.000 | 1.837 | 1.478 | 1.755 | 4.152 |
2 | 0.080 | 3.348 | 6.395 | 11.838 | 79.050 | 9.366 | 110.425 | 3.287 | 15.483 |
Dist. | ||||||
---|---|---|---|---|---|---|
NGE3E | 2.32272 | 5.56385 | 0.10187 | 0.31452 | – | – |
(0.49245) | (7.00899) | (1.27805) | (0.18822) | |||
EWE | 2.29855 | 0.04199 | – | – | 21.89576 | 1.04688 |
(1.21005) | (0.02975) | (19.11364) | (0.25627) | |||
WE | – | 0.03282 | – | – | 59.89789 | 1.54046 |
(0.04996) | (162.26158) | (0.16146) | ||||
EEW | – | 1.00906 | – | 1.15666 | 2.03844 | 1.33014 |
(31.52095) | (0.27759) | (0.99803) | (35.93074) | |||
BEF | – | 0.65072 | 91.12759 | 0.40903 | 13.59632 | 5.60806 |
(0.36349) | (41.07691) | (0.06542) | (52.52073) | (21.90342) | ||
OGEW | – | 21.13824 | 0.03953 | – | 2.42230 | 1.02483 |
(38.08836) | (0.08007) | (1.41950) | (0.32531) | |||
OGEFr | – | 43.95769 | 45.57000 | – | 0.68169 | 0.44654 |
(31.45097) | (32.38208) | (0.35684) | (0.08105) | |||
EKE | – | 16.65656 | 1.76429 | – | 0.08030 | 1.29005 |
(65.36980) | (1.20989) | (0.28969) | (0.58900) | |||
KwW | 0.86432 | – | – | 0.53177 | 2.73100 | 3.21403 |
(0.82088) | (0.46183) | (3.05683) | (9.65113) | |||
WL | 21.27763 | – | – | 0.70534 | 0.25353 | 2.19835 |
(55.95618) | (0.84875) | (0.22020) | (0.54623) |
Dist. | ||||||
---|---|---|---|---|---|---|
NGE3E | 1.46988 | 3.17280 | 0.04142 | 0.06681 | – | – |
(0.22276) | (1.65586) | (0.76421) | (0.04233) | |||
EWE | 4.03707 | 0.01171 | – | – | 7.07212 | 0.52346 |
(2.10918) | (0.00599) | (2.37210) | (0.11822) | |||
WE | – | 0.00691 | – | – | 13.77130 | 0.98836 |
(0.00218) | (5.32372) | (0.06271) | ||||
EEW | – | 0.75644 | – | 0.65442 | 2.79598 | 2.18408 |
(41.90063) | (0.13484) | (1.26511) | (184.93870) | |||
BEF | – | 154.72736 | 92.01702 | 0.05378 | 153.63292 | 1.85484 |
(256.89845) | (426.07475) | (0.03714) | (249.12744) | (2.52944) | ||
OGEW | – | 12.71015 | 0.04898 | – | 3.74616 | 0.53393 |
(16.73725) | (0.07842) | (1.99784) | (0.15390) | |||
OGEFr | – | 0.03261 | 0.02360 | – | 2.53262 | 0.67782 |
(0.03068) | (0.02448) | (1.01072) | (0.12675) | |||
EKE | – | 5.10315 | 3.89071 | – | 0.01387 | 0.48930 |
(3.05943) | (3.16005) | (0.00977) | (0.23850) | |||
KwW | 0.46255 | – | – | 0.49332 | 4.08136 | 2.87041 |
(0.39433) | (0.42793) | (4.52386) | (6.00747) | |||
WL | 14.27567 | – | – | 0.17142 | 0.17497 | 1.52178 |
(26.47401) | (0.16655) | (0.15495) | (0.28860) |
Dist. | -L | AIC | CAIC | BIC | KS | p-Value | CM | AD |
---|---|---|---|---|---|---|---|---|
NGE3E | 102.134 | 212.267 | 212.864 | 221.374 | 0.095 | 0.54 | 0.08360 | 0.57473 |
EWE | 102.976 | 213.952 | 214.548 | 223.058 | 0.101 | 0.4569 | 0.11519 | 0.75801 |
WE | 104.486 | 214.972 | 215.325 | 221.802 | 0.119 | 0.2632 | 0.17274 | 1.04466 |
EEW | 102.810 | 213.620 | 214.217 | 222.726 | 0.101 | 0.4499 | 0.11008 | 0.72722 |
BEF | 105.026 | 220.052 | 220.962 | 231.436 | 0.521 | 0.0000 | 0.12084 | 0.76687 |
OGEW | 102.979 | 213.959 | 214.556 | 223.065 | 0.102 | 0.4421 | 0.11361 | 0.75265 |
OGEFr | 105.679 | 219.353 | 219.950 | 228.460 | 0.131 | 0.1674 | 0.15307 | 1.07003 |
EKE | 102.764 | 213.528 | 214.125 | 222.635 | 0.100 | 0.4704 | 0.11084 | 0.72641 |
KwW | 102.710 | 213.420 | 214.017 | 222.526 | 0.101 | 0.4559 | 0.10771 | 0.71078 |
WL | 102.781 | 213.562 | 214.159 | 222.668 | 0.105 | 0.4045 | 0.11242 | 0.73235 |
Dist. | -L | AIC | CAIC | BIC | KS | p-Value | CM | AD |
---|---|---|---|---|---|---|---|---|
NGE3E | 409.823 | 827.645 | 827.971 | 839.053 | 0.037 | 0.9954 | 0.02147 | 0.14416 |
EWE | 411.320 | 830.641 | 830.966 | 842.049 | 0.048 | 0.9244 | 0.05658 | 0.37454 |
WE | 415.905 | 837.811 | 838.004 | 846.367 | 0.080 | 0.3840 | 0.16541 | 0.98962 |
EEW | 410.680 | 829.360 | 829.685 | 840.768 | 0.045 | 0.9576 | 0.04367 | 0.28848 |
BEF | 415.946 | 841.892 | 842.384 | 856.152 | 0.987 | 0.0000 | 0.20020 | 1.48189 |
OGEW | 410.922 | 829.844 | 830.169 | 841.252 | 0.047 | 0.9427 | 0.04845 | 0.32110 |
OGEFr | 410.838 | 829.676 | 830.001 | 841.084 | 0.044 | 0.9613 | 0.04767 | 0.31411 |
EKE | 411.000 | 830.000 | 830.325 | 841.408 | 0.046 | 0.9459 | 0.05222 | 0.33891 |
KwW | 410.569 | 829.138 | 829.464 | 840.546 | 0.045 | 0.9600 | 0.04145 | 0.27310 |
WL | 410.165 | 828.329 | 828.655 | 839.737 | 0.043 | 0.9721 | 0.03296 | 0.21647 |
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Hussain, S.; Sajid Rashid, M.; Ul Hassan, M.; Ahmed, R. The Generalized Exponential Extended Exponentiated Family of Distributions: Theory, Properties, and Applications. Mathematics 2022, 10, 3419. https://doi.org/10.3390/math10193419
Hussain S, Sajid Rashid M, Ul Hassan M, Ahmed R. The Generalized Exponential Extended Exponentiated Family of Distributions: Theory, Properties, and Applications. Mathematics. 2022; 10(19):3419. https://doi.org/10.3390/math10193419
Chicago/Turabian StyleHussain, Sajid, Muhammad Sajid Rashid, Mahmood Ul Hassan, and Rashid Ahmed. 2022. "The Generalized Exponential Extended Exponentiated Family of Distributions: Theory, Properties, and Applications" Mathematics 10, no. 19: 3419. https://doi.org/10.3390/math10193419
APA StyleHussain, S., Sajid Rashid, M., Ul Hassan, M., & Ahmed, R. (2022). The Generalized Exponential Extended Exponentiated Family of Distributions: Theory, Properties, and Applications. Mathematics, 10(19), 3419. https://doi.org/10.3390/math10193419