The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science
<p>Plots of the probability density function (PDF) and hazard rate function (HRF) of the exponential transmuted Burr X-G (TBXE) distribution for some parameter values.</p> "> Figure 2
<p>Plots of the PDF and HRF of the TBX-log-logistic (TBXLL) distribution for some parameter values.</p> "> Figure 3
<p>Fitted densities and distribution functions of the competing models for the first data set.</p> "> Figure 4
<p>Fitted densities and distribution functions of the competing models for the second data set.</p> "> Figure 5
<p>The hazard rate function (HRF) plot of the TBXE model and total time on test (TTT) plot for the first data set.</p> "> Figure 6
<p>The HRF plot of the TBXE model and TTT plot for the second data set.</p> "> Figure 7
<p>The probability–probability (PP) plot of the TBXE distribution and other fitted distributions for the first data set.</p> "> Figure 8
<p>The probability–probability (PP) plot of the TBXE distribution and other fitted distributions for the second data set.</p> ">
Abstract
:1. Introduction
2. Two Sub-Models
2.1. The TBXE Distribution
2.2. The TBXLL Distribution
3. Properties of the TBX-G Class
3.1. Useful Expansion for the TBX-G Density
3.2. Quantile Function
3.3. Moments
3.4. Residual and Reversed Residual Life Functions
3.5. Order Statistics
4. Properties of TBXE Distribution
4.1. Linear Representation
4.2. QF, Moments, and MGF
4.3. Mean Residual Life and Mean Inactivity Time
5. Maximum Likelihood Estimation
6. Eight Estimation Methods for TBXE Parameters
6.1. Maximum Likelihood
6.2. Anderson–Darling and Right-Tail Anderson–Darling
6.3. Cramér-Von Mises
6.4. Ordinary and Weighted Least-Squares
6.5. Maximum Product of Spacing
6.6. Percentile
7. Simulations
- All estimator methods showed consistency, except the MLE estimator method, which showed consistency for all parameter combinations except for combinations and .
- Form Table 1 and for the parameter combinations, we conclude that the ADE method outperformed all the other estimator methods (overall score of 71.5). Therefore, based on our study, we can consider the ADE method as the best.
8. Modeling Two Real Data
9. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
n | Est. | Est. Par. | WLSE | OLSE | MLE | MPSE | CVME | ADE | RADE | PCE |
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n | WLSE | OLSE | MLE | MPSE | CVME | ADE | RADE | PCE | |
---|---|---|---|---|---|---|---|---|---|
50 | 5 | 7 | 5 | 2 | 8 | 3 | 5 | 1 | |
150 | 5 | 7 | 6 | 2 | 8 | 3 | 4 | 1 | |
300 | 6 | 7.5 | 4.5 | 1 | 7.5 | 3 | 4.5 | 2 | |
400 | 6 | 7 | 5 | 1 | 8 | 4 | 3 | 2 | |
50 | 5 | 7 | 4 | 8 | 6 | 2 | 1 | 3 | |
150 | 2 | 7 | 6 | 8 | 5 | 1 | 4 | 3 | |
300 | 2 | 8 | 6.5 | 6.5 | 5 | 1 | 4 | 3 | |
400 | 2 | 8 | 6 | 7 | 5 | 1 | 4 | 3 | |
50 | 2 | 5.5 | 3 | 7 | 4 | 1 | 8 | 5.5 | |
150 | 2 | 5.5 | 5.5 | 1 | 4 | 3 | 7 | 8 | |
300 | 3 | 5 | 4 | 1 | 6 | 2 | 7 | 8 | |
400 | 2 | 5 | 3 | 1 | 7 | 4 | 8 | 6 | |
50 | 2 | 6 | 7 | 8 | 5 | 1 | 4 | 3 | |
150 | 2 | 5 | 7 | 3 | 6 | 1 | 4 | 8 | |
300 | 2 | 7 | 6 | 1 | 5 | 3 | 4 | 8 | |
400 | 1.5 | 4 | 7 | 1.5 | 6 | 3 | 5 | 8 | |
50 | 5 | 6 | 1 | 8 | 4 | 2 | 3 | 7 | |
150 | 3 | 7 | 1 | 8 | 4 | 2 | 5 | 6 | |
300 | 3 | 8 | 1 | 7 | 6 | 2 | 5 | 4 | |
400 | 3 | 8 | 1 | 6 | 7 | 2 | 5 | 4 | |
50 | 5 | 6 | 7 | 2 | 8 | 3 | 4 | 1 | |
150 | 5 | 6 | 8 | 1 | 7 | 4 | 3 | 2 | |
300 | 5 | 6 | 8 | 1 | 7 | 3 | 4 | 2 | |
400 | 5 | 6 | 8 | 1 | 7 | 3 | 4 | 2 | |
50 | 2 | 6 | 3 | 7 | 5 | 1 | 8 | 4 | |
150 | 1 | 5 | 6 | 3.5 | 3.5 | 2 | 7 | 8 | |
300 | 2 | 5 | 4 | 1 | 6 | 3 | 8 | 7 | |
400 | 2 | 6.5 | 4 | 1 | 6.5 | 3 | 8 | 5 | |
50 | 2 | 4.5 | 7 | 6 | 4.5 | 1 | 3 | 8 | |
150 | 1.5 | 4 | 7 | 6 | 5 | 1.5 | 3 | 8 | |
300 | 2 | 4 | 6.5 | 3 | 5 | 1 | 6.5 | 8 | |
400 | 3 | 7 | 5 | 1 | 6 | 2 | 4 | 8 | |
99 | 196.5 | 163 | 121.5 | 187 | 71.5 | 157 | 156.5 | ||
2 | 8 | 6 | 3 | 7 | 1 | 5 | 4 |
n | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|
100 | 0.390 | 1.840 | 2.7 | 2.621 | 3.22 | 5.56 | 0.3682 | 3.10494 |
n | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|
40 | 1.6 | 5.075 | 6.5 | 6.253 | 7.825 | 9 | 2.641 |
Distribution | Estimates | SEs | AIC | CAIC | BIC | HQIC | W | A | K-S (stat) | K-S (p-Value) | |
TBXE | 0.0504 | 141.4421 | 288.8843 | 289.1343 | 296.6998 | 292.0473 | 0.0558 | 0.3942 | 0.0556 | 0.9167 | |
0.2481 | |||||||||||
0.8499 | |||||||||||
MOLE | 2.7204 | 141.8820 | 289.7640 | 290.0140 | 297.5795 | 292.9270 | 0.0646 | 0.3974 | 0.0602 | 0.8611 | |
2.1449 | |||||||||||
110.6220 | |||||||||||
Ga | 1.6289 | 143.3214 | 290.6428 | 290.7665 | 295.8531 | 292.7515 | 0.1483 | 0.7586 | 0.0786 | 0.5676 | |
0.6393 | |||||||||||
BE | 1.6852 | 143.3920 | 292.7840 | 293.0340 | 300.5995 | 295.9471 | 0.1503 | 0.7692 | 0.0788 | 0.5642 | |
35.3987 | |||||||||||
0.3655 | |||||||||||
GTPE | 0.1614 | 143.3149 | 292.6298 | 292.8798 | 300.4453 | 295.7929 | 0.1543 | 0.7775 | 0.0800 | 0.5448 | |
0.8771 | |||||||||||
6.7468 | |||||||||||
APE | 48,068.1765 | 17,441.3621 | 144.2753 | 292.5505 | 292.6743 | 297.7609 | 294.6593 | 0.1857 | 0.9472 | 0.0830 | 0.4967 |
0.0491 | |||||||||||
TGE | 5.2757 | 144.5828 | 295.1655 | 295.4155 | 302.9810 | 298.3286 | 0.1867 | 0.9565 | 0.0830 | 0.4959 | |
0.1741 | |||||||||||
0.7526 | |||||||||||
TEGE | 28.4292 | 144.5828 | 297.1656 | 297.5867 | 307.5863 | 301.3831 | 0.1867 | 0.9565 | 0.0830 | 0.4959 | |
5.2757 | |||||||||||
0.7527 | |||||||||||
5.1732 | |||||||||||
EE | 3.4978 | 146.5126 | 297.0252 | 297.1489 | 302.2355 | 299.1339 | 0.2349 | 1.2320 | 0.0875 | 0.4285 | |
0.1598 | |||||||||||
FWME | 277.2217 | 146.3190 | 300.6381 | 301.0592 | 311.0588 | 304.8555 | 0.2395 | 1.2437 | 0.0890 | 0.4064 | |
420.0028 | |||||||||||
110.6217 | |||||||||||
230.2469 | |||||||||||
E | 0.0307 | 199.3820 | 400.7641 | 400.8049 | 403.3693 | 401.8184 | 0.1542 | 0.7904 | 0.2504 |
Distribution | Estimates | SEs | AIC | CAIC | BIC | HQIC | W | A | K-S(stat) | K-S(p-Value) | |
---|---|---|---|---|---|---|---|---|---|---|---|
TBXE | 0.0108 | 81.9019 | 169.8038 | 170.4704 | 174.8704 | 171.6357 | 0.0633 | 0.4753 | 0.0832 | 0.9445 | |
1.0054 | |||||||||||
1.0565 | |||||||||||
MOLE | 21.8544 | 83.8204 | 173.6409 | 174.3075 | 178.7075 | 175.4728 | 0.0838 | 0.6036 | 0.1007 | 0.8117 | |
3.6142 | |||||||||||
289.7554 | |||||||||||
Ga | 3.7431 | 87.6709 | 179.3418 | 179.6661 | 182.7196 | 180.5631 | 0.2058 | 1.3647 | 0.1331 | 0.4777 | |
0.5896 | |||||||||||
BE | 3.5522 | 87.6205 | 181.2410 | 181.9077 | 186.3076 | 183.0729 | 0.2059 | 1.3655 | 0.1387 | 0.4247 | |
53.7494 | |||||||||||
0.0119 | |||||||||||
GTPE | 0.1148 | 87.9930 | 181.9860 | 182.6527 | 187.0526 | 183.8179 | 0.2057 | 1.3605 | 0.1486 | 0.3399 | |
0.7033 | |||||||||||
10.4039 | |||||||||||
APE | 318,294.3557 | 16,781.3618 | 89.0895 | 182.1790 | 182.5033 | 185.5567 | 183.4003 | 0.2336 | 1.5244 | 0.1666 | 0.2170 |
0.0295 | |||||||||||
TGE | 9.1154 | 88.8872 | 183.7743 | 184.4410 | 188.8410 | 185.6063 | 0.2363 | 1.5353 | 0.1520 | 0.3136 | |
0.1225 | |||||||||||
0.6411 | |||||||||||
TEGE | 1.3161 | 88.8872 | 185.7745 | 186.9174 | 192.5300 | 188.2171 | 0.2363 | 1.5353 | 0.1520 | 0.3137 | |
9.1169 | |||||||||||
0.6412 | |||||||||||
17.9404 | |||||||||||
EE | 8.3549 | 90.6454 | 185.2909 | 185.6152 | 188.6686 | 186.5122 | 0.2882 | 1.8287 | 0.1631 | 0.2377 | |
0.1118 | |||||||||||
FWME | 1.8998 | 88.7075 | 185.4150 | 186.5579 | 192.1705 | 187.8576 | 0.2147 | 1.4101 | 0.1448 | 0.3715 | |
378.1227 | |||||||||||
13.9570 | |||||||||||
213.1556 | |||||||||||
E | 0.0197 | 114.7675 | 231.5350 | 231.6402 | 233.2238 | 232.1456 | 0.2111 | 1.3954 | 0.3376 | 0.0002 |
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Al-Babtain, A.A.; Elbatal, I.; Al-Mofleh, H.; Gemeay, A.M.; Afify, A.Z.; Sarg, A.M. The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science. Symmetry 2021, 13, 474. https://doi.org/10.3390/sym13030474
Al-Babtain AA, Elbatal I, Al-Mofleh H, Gemeay AM, Afify AZ, Sarg AM. The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science. Symmetry. 2021; 13(3):474. https://doi.org/10.3390/sym13030474
Chicago/Turabian StyleAl-Babtain, Abdulhakim A., Ibrahim Elbatal, Hazem Al-Mofleh, Ahmed M. Gemeay, Ahmed Z. Afify, and Abdullah M. Sarg. 2021. "The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science" Symmetry 13, no. 3: 474. https://doi.org/10.3390/sym13030474