Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution
Abstract
:1. Introduction
2. The Type I Half-Logistic Topp–Leone Distribution
3. Mathematical Properties
3.1. Asymptotes and Shapes
3.2. Quantile Function
3.3. Skewness and Kurtosis
3.4. Power Series Expansion
3.5. Ordinary Moments
3.6. Incomplete Moments
3.7. Moment-Generating Function
3.8. Stress Strength Parameter
3.9. Order Statistics
4. Parameter Estimation
4.1. Method of Maximum Likelihood Estimation
4.2. Methods of Least Squares and Weighted Least Squares Estimation
4.3. Method of Cramer–von Mises Minimum Distance Estimation
4.4. Method of Percentile Estimation
4.5. Methods of Anderson–Darling and Right-Tail Anderson–Darling Estimation
4.6. Simulation
5. Applications
- The Kumaraswamy (Kum) model with pdf given by
- The Topp–Leone (TL) model with pdf given by
- The beta (B) model with pdf given by
- The Topp–Leone exponential (TLE) model with pdf given by
6. Concluding Remarks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 2.172 | 0.628 | 2.038 | 0.473 | 2.058 | 0.508 | 1.808 | 0.583 | 2.148 | 3.377 | 2.067 | 2.974 | 2.194 | 3.811 |
0.507 | 7.116 * | 0.499 | 5.885 * | 0.498 | 6.713 * | 0.472 | 0.013 | 0.499 | 6.241 * | 0.499 | 6.682 * | 0.498 | 7.554 * | |
100 | 2.164 | 0.271 | 2.011 | 0.206 | 2.062 | 0.340 | 1.852 | 0.272 | 2.159 | 3.139 | 2.064 | 2.766 | 2.131 | 3.288 |
0.515 | 2.525 * | 0.499 | 2.756 * | 0.495 | 2.766 * | 0.473 | 7.531 * | 0.515 | 3.873 * | 0.496 | 2.697 * | 0.501 | 3.809 * | |
200 | 2.106 | 0.098 | 2.023 | 0.157 | 2.029 | 0.107 | 1.912 | 0.130 | 2.057 | 2.587 | 2.015 | 2.379 | 2.030 | 2.532 |
0.503 | 1.184 * | 0.500 | 1.891 * | 0.502 | 1.514 * | 0.488 | 4.459 * | 0.501 | 1.827 * | 0.501 | 1.220 * | 0.501 | 1.848 * | |
1000 | 1.997 | 0.057 | 2.012 | 0.024 | 2.012 | 0.019 | 1.998 | 0.039 | 2.020 | 2.334 | 2.010 | 2.298 | 2.027 | 2.365 |
0.499 | 1.381 * | 0.501 | 0.313 * | 0.501 | 0.247 * | 0.503 | 1.741 * | 0.502 | 0.317 * | 0.501 | 0.251 * | 0.502 | 0.324 * |
MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 2.136 | 0.213 | 1.949 | 0.188 | 2.049 | 0.210 | 1.794 | 0.272 | 2.092 | 0.614 | 2.043 | 0.456 | 2.130 | 0.679 |
1.593 | 0.106 | 1.465 | 0.064 | 1.538 | 0.114 | 1.350 | 0.153 | 1.588 | 0.135 | 1.552 | 0.081 | 1.573 | 0.136 | |
100 | 2.063 | 0.077 | 2.051 | 0.130 | 2.008 | 0.118 | 1.780 | 0.181 | 2.079 | 0.438 | 2.012 | 0.346 | 2.069 | 0.470 |
1.535 | 0.033 | 1.539 | 0.051 | 1.502 | 0.046 | 1.333 | 0.167 | 1.565 | 0.050 | 1.504 | 0.030 | 1.535 | 0.050 | |
200 | 2.026 | 0.047 | 2.016 | 0.057 | 2.021 | 0.047 | 1.850 | 0.085 | 2.055 | 0.370 | 2.011 | 0.312 | 2.042 | 0.340 |
1.529 | 0.018 | 1.516 | 0.026 | 1.528 | 0.022 | 1.387 | 0.069 | 1.543 | 0.030 | 1.503 | 0.022 | 1.540 | 0.021 | |
1000 | 2.009 | 7.847 * | 2.007 | 0.010 | 2.008 | 8.377 * | 1.961 | 0.016 | 2.012 | 0.272 | 2.006 | 0.265 | 2.014 | 0.276 |
1.507 | 3.286 * | 1.505 | 4.59 *1 | 1.506 | 3.595 * | 1.465 | 0.013 | 1.509 | 4.675 * | 1.505 | 3.699 * | 1.510 | 4.574 * |
MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 2.065 | 0.164 | 2.034 | 0.199 | 2.060 | 0.204 | 1.781 | 0.274 | 2.104 | 0.238 | 2.012 | 0.157 | 2.078 | 0.230 |
2.089 | 0.161 | 2.047 | 0.230 | 2.071 | 0.226 | 1.790 | 0.474 | 2.144 | 0.332 | 2.042 | 0.153 | 2.105 | 0.255 | |
100 | 2.028 | 0.071 | 1.984 | 0.083 | 2.035 | 0.105 | 1.861 | 0.139 | 2.058 | 0.092 | 2.032 | 0.071 | 2.047 | 0.101 |
2.033 | 0.088 | 2.002 | 0.084 | 2.068 | 0.122 | 1.856 | 0.212 | 2.094 | 0.122 | 2.051 | 0.075 | 2.074 | 0.102 | |
200 | 2.023 | 0.043 | 2.014 | 0.051 | 2.014 | 0.034 | 1.932 | 0.060 | 2.016 | 0.052 | 2.026 | 0.038 | 1.995 | 0.049 |
2.038 | 0.043 | 2.005 | 0.063 | 2.017 | 0.031 | 1.920 | 0.117 | 2.013 | 0.067 | 2.037 | 0.038 | 1.993 | 0.039 | |
1000 | 2.008 | 6.831 * | 2.007 | 8.687 * | 2.008 | 7.203 * | 1.966 | 0.014 | 2.011 | 8.796 * | 2.006 | 7.359 * | 2.012 | 0.010 |
2.010 | 6.858 * | 2.008 | 9.561 * | 2.010 | 7.461 * | 1.955 | 0.028 | 2.013 | 9.735 * | 2.008 | 7.691 * | 2.014 | 9.411 * |
MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 3.392 | 1.117 | 3.173 | 1.883 | 3.285 | 1.993 | 2.624 | 1.241 | 3.460 | 10.441 | 3.183 | 8.504 | 3.435 | 11.453 |
0.521 | 6.864 * | 0.505 | 7.689 * | 0.514 | 9.365 * | 0.470 | 0.016 | 0.523 | 8.203 * | 0.507 | 6.952 * | 0.513 | 7.557 * | |
100 | 3.163 | 0.486 | 3.186 | 0.794 | 3.043 | 0.579 | 2.720 | 0.735 | 3.373 | 9.084 | 3.102 | 7.179 | 3.163 | 7.792 |
0.507 | 2.750 * | 0.506 | 4.799 * | 0.502 | 2.578 * | 0.470 | 8.690 * | 0.518 | 5.421 * | 0.508 | 2.955 * | 0.505 | 2.814 * | |
200 | 3.047 | 0.204 | 3.121 | 0.485 | 3.059 | 0.208 | 2.890 | 0.357 | 3.133 | 7.383 | 3.027 | 6.630 | 3.159 | 7.470 |
0.506 | 1.541 * | 0.505 | 1.617 * | 0.503 | 0.950 * | 0.478 | 5.915 * | 0.506 | 1.689 * | 0.499 | 1.462 * | 0.503 | 1.766 * | |
1000 | 3.026 | 0.040 | 3.018 | 0.054 | 3.018 | 0.042 | 2.950 | 0.064 | 3.030 | 6.453 | 3.015 | 6.366 | 3.041 | 6.529 |
0.502 | 0.227 * | 0.501 | 0.313 * | 0.501 | 0.246 * | 0.496 | 0.976 * | 0.502 | 0.317 * | 0.501 | 0.251 * | 0.502 | 0.324 * |
MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 3.008 | 0.367 | 3.058 | 0.490 | 3.070 | 0.463 | 2.771 | 0.654 | 3.224 | 3.577 | 3.105 | 2.921 | 3.049 | 2.902 |
1.567 | 0.090 | 1.551 | 0.110 | 1.495 | 0.069 | 1.451 | 0.282 | 1.639 | 0.149 | 1.538 | 0.068 | 1.518 | 0.078 | |
100 | 3.190 | 0.256 | 3.103 | 0.350 | 3.111 | 0.206 | 2.753 | 0.323 | 3.191 | 3.182 | 3.016 | 2.522 | 3.020 | 2.630 |
1.582 | 0.059 | 1.554 | 0.076 | 1.536 | 0.041 | 1.374 | 0.127 | 1.601 | 0.089 | 1.506 | 0.036 | 1.518 | 0.058 | |
200 | 3.050 | 0.081 | 2.946 | 0.157 | 3.073 | 0.144 | 2.777 | 0.199 | 3.001 | 2.408 | 3.002 | 2.379 | 3.034 | 2.476 |
1.530 | 0.018 | 1.485 | 0.030 | 1.523 | 0.027 | 1.378 | 0.069 | 1.510 | 0.033 | 1.498 | 0.023 | 1.521 | 0.023 | |
1000 | 3.014 | 0.018 | 3.011 | 0.023 | 3.014 | 0.019 | 2.942 | 0.036 | 3.018 | 2.329 | 3.010 | 2.298 | 3.021 | 2.340 |
1.507 | 3.265 * | 1.505 | 4.599 * | 1.507 | 3.676 * | 1.465 | 0.013 | 1.509 | 4.686 * | 1.505 | 3.698 * | 1.510 | 4.575 * |
MLE | LSE | WLSE | PCE | CVE | ADE | RTADE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs | Est. | MSEs |
50 | 3.165 | 0.341 | 2.996 | 0.360 | 3.003 | 0.335 | 2.723 | 0.747 | 3.125 | 1.686 | 3.040 | 1.500 | 3.109 | 1.676 |
2.101 | 0.204 | 2.001 | 0.183 | 2.049 | 0.154 | 1.861 | 0.448 | 2.125 | 0.227 | 2.093 | 0.200 | 2.081 | 0.212 | |
100 | 3.085 | 0.177 | 3.022 | 0.226 | 3.113 | 0.209 | 2.692 | 0.317 | 3.048 | 1.318 | 3.029 | 1.253 | 3.059 | 1.324 |
2.048 | 0.076 | 2.045 | 0.173 | 2.099 | 0.110 | 1.765 | 0.239 | 2.054 | 0.147 | 2.030 | 0.088 | 2.037 | 0.086 | |
200 | 2.993 | 0.073 | 3.032 | 0.100 | 2.988 | 0.101 | 2.825 | 0.193 | 3.102 | 1.321 | 3.025 | 1.118 | 3.004 | 1.145 |
1.998 | 0.036 | 2.031 | 0.045 | 1.997 | 0.048 | 1.866 | 0.128 | 2.078 | 0.060 | 2.015 | 0.034 | 2.030 | 0.062 | |
1000 | 3.012 | 0.015 | 3.010 | 0.020 | 3.010 | 0.016 | 2.942 | 0.034 | 3.017 | 1.054 | 3.009 | 1.034 | 3.018 | 1.060 |
2.010 | 6.858 * | 2.008 | 9.564 * | 2.009 | 7.425 * | 1.947 | 0.029 | 2.013 | 9.728 * | 2.008 | 7.689 * | 2.014 | 9.413 * |
Parameters | n | MLE | LSE | WLSE | PCE | CVE | ADE | RTADE |
---|---|---|---|---|---|---|---|---|
, | 50 | 4.5 | 1.0 | 2.0 | 6.0 | 3.0 | 4.5 | 7.0 |
100 | 1.0 | 2.0 | 3.5 | 5.0 | 6.5 | 4.5 | 6.5 | |
200 | 1.0 | 4.5 | 2.0 | 4.5 | 6.5 | 3.0 | 6.5 | |
1000 | 5.5 | 2.0 | 1.0 | 5.5 | 3.5 | 3.5 | 7.0 | |
, | 50 | 2.5 | 1.0 | 2.5 | 5.0 | 6.5 | 4.0 | 6.5 |
100 | 1.0 | 4.0 | 2.0 | 6.0 | 5.0 | 3.0 | 7.0 | |
200 | 1.0 | 3.0 | 2.0 | 6.0 | 7.0 | 5.0 | 4.0 | |
1000 | 1.0 | 3.5 | 2.0 | 5.5 | 7.0 | 3.5 | 5.5 | |
, | 50 | 2.0 | 3.5 | 3.5 | 7.0 | 6.0 | 1.0 | 5.0 |
100 | 2.0 | 3.0 | 6.0 | 7.0 | 5.0 | 1.0 | 4.0 | |
200 | 3.5 | 5.0 | 1.0 | 7.0 | 6.0 | 2.0 | 3.5 | |
1000 | 1.0 | 4.0 | 2.0 | 7.0 | 5.0 | 7.0 | 6.0 | |
, | 50 | 1.0 | 2.5 | 5.5 | 4.0 | 7.0 | 2.5 | 5.5 |
100 | 1.5 | 3.0 | 1.5 | 6.0 | 7.0 | 3.0 | 3.0 | |
200 | 2.0 | 4.0 | 1.0 | 5.0 | 6.0 | 3.0 | 7.0 | |
1000 | 1.0 | 3.0 | 2.0 | 5.5 | 5.5 | 4.0 | 7.0 | |
, | 50 | 1.5 | 4.5 | 1.5 | 6.0 | 7.0 | 3.0 | 4.5 |
100 | 2.5 | 4.5 | 1.0 | 6.0 | 7.0 | 2.5 | 4.5 | |
200 | 1.0 | 4.0 | 2.0 | 6.0 | 7.0 | 3.0 | 5.0 | |
1000 | 1.0 | 3.5 | 2.0 | 5.5 | 7.0 | 3.5 | 5.5 | |
, | 50 | 2.5 | 1.0 | 2.5 | 7.0 | 6.0 | 4.0 | 5.0 |
100 | 1.5 | 5.5 | 3.0 | 7.5 | 1.5 | 7.5 | 4.0 | |
200 | 1.0 | 2.5 | 4.0 | 5.5 | 5.5 | 2.5 | 7.0 | |
1000 | 1.0 | 3.5 | 2.0 | 5.5 | 7.0 | 3.5 | 5.5 | |
Sum of ranks | 43.5 | 78 | 57.5 | 141 | 140.5 | 84 | 132 | |
Overall rank | 1 | 3 | 2 | 7 | 6 | 4 | 5 |
n | Mean | Median | Standard Deviation | Skewness | Kurtosis | |
---|---|---|---|---|---|---|
Data set 1 | 20 | 0.16 | 0.13 | 0.16 | 1.23 | 1.07 |
Data set 2 | 50 | 0.16 | 0.16 | 0.08 | 0.07 | −0.87 |
Model | λ | α | a | b |
---|---|---|---|---|
TIHLTL | 2.1342 | 0.6028 | - | - |
(0.6336) | (0.1706) | - | - | |
Kum | - | - | 0.7639 | 3.4341 |
- | - | (0.1751) | (1.3113) | |
TL | - | 0.5112 | - | - |
- | (0.1143) | - | - | |
B | - | - | 0.7134 | 3.7459 |
- | - | (0.1931) | ( 1.3085) | |
TLE | 0.7925 | 2.6644 | - | - |
(0.2208) | (0.8240) | - | - |
Model | λ | α | a | b |
---|---|---|---|---|
TIHLTL | 10.6094 | 1.8752 | - | - |
(1.1160) | (0.2643) | - | - | |
Kum | - | - | 2.0750 | 33.0041 |
- | - | (0.2542) | (3.8351) | |
TL | - | 0.7247 | - | - |
- | (0.1024) | - | - | |
B | - | - | 2.6799 | 13.8502 |
- | - | (0.5066) | (2.8249) | |
TLE | 3.1718 | 5.6800 | - | - |
(0.7067) | (0.7841) | - | - |
Model | AIC | BIC | W* | A* | KS | p-Value (KS) | |
---|---|---|---|---|---|---|---|
TIHLTL | −17.3028 | −30.6057 | −28.6143 | 0.0208 | 0.1348 | 0.0900 | 0.9912 |
Kum | −17.2047 | −30.4094 | −28.4180 | 0.0289 | 0.1691 | 0.1026 | 0.9700 |
TL | −15.6166 | −29.2333 | −28.2376 | 0.0300 | 0.1754 | 0.1848 | 0.4481 |
B | −17.2532 | −30.5064 | −28.5150 | 0.0263 | 0.1565 | 0.0980 | 0.9782 |
TLE | −16.8608 | −29.7216 | −27.7301 | 0.0375 | 0.2124 | 0.1225 | 0.8899 |
Model | AIC | BIC | W* | A* | KS | p-Value (KS) | |
---|---|---|---|---|---|---|---|
TIHLTL | −56.4261 | −108.8522 | −105.0282 | 0.0893 | 0.5496 | 0.1031 | 0.6623 |
Kum | −56.0686 | −108.1373 | −104.3132 | 0.1024 | 0.6248 | 0.1104 | 0.5755 |
TL | −28.4078 | −54.8156 | −52.9035 | 0.1653 | 0.9919 | 0.3622 | 0.000003 |
B | −54.6066 | −105.2133 | −101.3892 | 0.1479 | 0.8926 | 0.1414 | 0.2697 |
TLE | −52.2862 | −100.5725 | −96.7484 | 0.2120 | 1.2634 | 0.1652 | 0.1302 |
CI | λ | α |
95% | [0.8923 3.3760] | [0.2684 0.9371] |
99% | [0.4995 3.7688] | [0.1626 1.0429] |
CI | λ | α |
95% | [8.4220 12.7967] | [1.3571 2.3932] |
99% | [7.7301 13.4886] | [1.1933 2.5570] |
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ZeinEldin, R.A.; Chesneau, C.; Jamal, F.; Elgarhy, M. Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution. Mathematics 2019, 7, 985. https://doi.org/10.3390/math7100985
ZeinEldin RA, Chesneau C, Jamal F, Elgarhy M. Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution. Mathematics. 2019; 7(10):985. https://doi.org/10.3390/math7100985
Chicago/Turabian StyleZeinEldin, Ramadan A., Christophe Chesneau, Farrukh Jamal, and Mohammed Elgarhy. 2019. "Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution" Mathematics 7, no. 10: 985. https://doi.org/10.3390/math7100985
APA StyleZeinEldin, R. A., Chesneau, C., Jamal, F., & Elgarhy, M. (2019). Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution. Mathematics, 7(10), 985. https://doi.org/10.3390/math7100985