Figure 1.
Plots of the Xg-BXII distribution pdf.
Figure 1.
Plots of the Xg-BXII distribution pdf.
Figure 2.
Plots of the Xg-BXII distribution hrf.
Figure 2.
Plots of the Xg-BXII distribution hrf.
Figure 3.
Plots of the Xg-BXII distribution rhrf.
Figure 3.
Plots of the Xg-BXII distribution rhrf.
Figure 4.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus .
Figure 4.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus .
Figure 5.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus n.
Figure 5.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus n.
Figure 6.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus n.
Figure 6.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the parameters, for versus n.
Figure 7.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf, for versus n.
Figure 7.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf, for versus n.
Figure 8.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf, for versus n.
Figure 8.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf, for versus n.
Figure 9.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf for versus n.
Figure 9.
Plots of the ML averages, estimated risks, relative errors, relative absolute biases, variances and the ACI lengths of the rf, hrf and rhrf for versus n.
Figure 10.
The empirical scaled TTT-transform, boxplot and histogram of Wang’s data.
Figure 10.
The empirical scaled TTT-transform, boxplot and histogram of Wang’s data.
Figure 11.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Wang’s data.
Figure 11.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Wang’s data.
Figure 12.
Histogram of Wang’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 12.
Histogram of Wang’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 13.
P-P plot of the competitor models for Wang’s data.
Figure 13.
P-P plot of the competitor models for Wang’s data.
Figure 14.
The empirical scaled TTT-transform, boxplot and histogram of Meeker and Escober’s data.
Figure 14.
The empirical scaled TTT-transform, boxplot and histogram of Meeker and Escober’s data.
Figure 15.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Meeker and Escober’s data.
Figure 15.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Meeker and Escober’s data.
Figure 16.
Histogram of Meeker and Escober’s data and the fitted pdfs and Empirical cdf versus the fitted cdfs of the competitor models.
Figure 16.
Histogram of Meeker and Escober’s data and the fitted pdfs and Empirical cdf versus the fitted cdfs of the competitor models.
Figure 17.
P-P plot of the competitor models for Meeker and Escober’s data.
Figure 17.
P-P plot of the competitor models for Meeker and Escober’s data.
Figure 18.
The empirical scaled TTT-transform, boxplot and histogram of Murthy’s data.
Figure 18.
The empirical scaled TTT-transform, boxplot and histogram of Murthy’s data.
Figure 19.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Murthy’s data.
Figure 19.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Murthy’s data.
Figure 20.
Histogram of Murthy’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 20.
Histogram of Murthy’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 21.
P-P plot of the competitor models for Murthy’s data.
Figure 21.
P-P plot of the competitor models for Murthy’s data.
Figure 22.
The empirical scaled TTT-transform, boxplot and histogram of the United Kingdom’s data.
Figure 22.
The empirical scaled TTT-transform, boxplot and histogram of the United Kingdom’s data.
Figure 23.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of the United Kingdom’s data.
Figure 23.
Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of the United Kingdom’s data.
Figure 24.
Histogram of the United Kingdom’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 24.
Histogram of the United Kingdom’s data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.
Figure 25.
P-P plot of the United Kingdom’s data.
Figure 25.
P-P plot of the United Kingdom’s data.
Table 1.
Some quartiles and the mode of the Xg-BXII distribution for different parameters values.
Table 1.
Some quartiles and the mode of the Xg-BXII distribution for different parameters values.
α | c | k | x0.25 | xm | x0.75 | Mode |
---|
2.6 | 5 | 6 | 0.1610 | 0.3807 | 0.5899 | 0.5563 |
2 | 0.5 | 0.5 | 0.1115 | 0.3748 | 0.9107 | - |
0.5 | 3 | 1.5 | 0.5275 | 0.7756 | 1.0800 | 0.6789 |
0.25 | 3 | 5 | 0.3810 | 0.5229 | 0.6775 | 0.4955 |
0.15 | 5 | 0.8 | 0.8346 | 1.0571 | 1.3498 | 0.9533 |
0.05 | 4 | 2 | 0.6263 | 0.8016 | 0.9994 | 0.7595 |
Table 2.
Moments of the Xg-BXII distribution for different parameters values.
Table 2.
Moments of the Xg-BXII distribution for different parameters values.
α | c | k | μ | | | | μ2 | CV | CS | CK |
---|
2.6 | 5 | 6 | 0.3878 | 0.2135 | 0.1352 | 0.0929 | 0.0630 | 0.6474 | 0.2193 | 2.0180 |
2 | 0.5 | 0.5 | 0.6330 | 0.9187 | 1.9520 | 5.3099 | 0.5181 | 1.1371 | 1.9165 | 7.8042 |
0.5 | 3 | 1.5 | 0.8529 | 0.9717 | 1.4616 | 3.0361 | 0.2444 | 0.5796 | 1.7888 | 11.7837 |
0.25 | 3 | 5 | 0.5391 | 0.3405 | 0.2430 | 0.1926 | 0.0499 | 0.4142 | 0.5131 | 3.5599 |
0.15 | 5 | 0.8 | 1.1585 | 1.6443 | 3.1835 | 10.8832 | 0.3022 | 0.4745 | 3.4822 | 43.457 |
0.05 | 4 | 2 | 0.8321 | 0.7841 | 0.8313 | 0.9974 | 0.0917 | 0.3639 | 0.9458 | 5.9130 |
Table 3.
Sub-models of the Xg-BXII distribution.
Table 3.
Sub-models of the Xg-BXII distribution.
Parameter | The Resulting Distribution | Pdf |
---|
c = 1 | Xg-compound exponential distribution | |
c = 2 | Xg-compound Rayleigh distribution | |
k = 1 | Xg-log logistic | |
| BXII distribution | |
and
| Lomax distribution | |
and
| Compound Rayleigh distribution | |
and | Log logistic distribution | |
| Xgamma distribution | |
Table 4.
ML averages, estimated risks, relative errors, Relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
Table 4.
ML averages, estimated risks, relative errors, Relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
n | | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 1.9502 | 0.1022 | 0.1599 | 0.0249 | 0.0998 | 2.5692 | 1.3311 | 1.2381 |
| 0.6009 | 0.0556 | 0.4717 | 0.2017 | 0.0455 | 1.0186 | 0.1830 | 0.8358 |
| 0.3736 | 0.0557 | 0.7865 | 0.2453 | 0.0503 | 0.8130 | 0 | 0.8130 |
60 | | 1.9442 | 0.0760 | 0.1378 | 0.0279 | 0.0728 | 2.4732 | 1.4152 | 1.0579 |
| 0.5770 | 0.0413 | 0.4065 | 0.1541 | 0.0354 | 0.9456 | 0.2084 | 0.7372 |
| 0.3806 | 0.0513 | 0.7552 | 0.2687 | 0.0448 | 0.7956 | 0 | 0.7956 |
100 | | 1.9443 | 0.0477 | 0.1092 | 0.0279 | 0.0446 | 2.3582 | 1.5304 | 0.8278 |
| 0.5518 | 0.0347 | 0.3724 | 0.1036 | 0.0320 | 0.9023 | 0.2013 | 0.7011 |
| 0.3742 | 0.0452 | 0.7090 | 0.2472 | 0.0397 | 0.7649 | 0 | 0.7649 |
200 | | 1.9418 | 0.0356 | 0.0943 | 0.0291 | 0.0322 | 2.2936 | 1.5900 | 0.7036 |
| 0.5395 | 0.0265 | 0.3253 | 0.0789 | 0.0249 | 0.8488 | 0.2302 | 0.6186 |
| 0.3637 | 0.0355 | 0.6276 | 0.2124 | 0.0314 | 0.7110 | 0.0164 | 0.6946 |
500 | | 1.9679 | 0.0176 | 0.0664 | 0.0160 | 0.0166 | 2.2204 | 1.7155 | 0.5049 |
| 0.5163 | 0.0136 | 0.2331 | 0.0326 | 0.0133 | 0.7425 | 0.2901 | 0.4524 |
| 0.3373 | 0.0199 | 0.4699 | 0.1244 | 0.0185 | 0.6038 | 0.0709 | 0.5329 |
Table 5.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
Table 5.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
n | | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 0.5110 | 0.1166 | 0.6830 | 0.0219 | 0.1165 | 1.1799 | 0 | 1.1799 |
| 3.1666 | 0.2807 | 0.1766 | 0.0555 | 0.2530 | 4.1524 | 2.1808 | 1.9716 |
| 1.4544 | 0.0955 | 0.2060 | 0.0304 | 0.0934 | 2.0534 | 0.8554 | 1.1981 |
60 | | 0.4926 | 0.0881 | 0.5935 | 0.0148 | 0.0880 | 1.0740 | 0 | 1.0740 |
| 3.1551 | 0.2316 | 0.1604 | 0.0517 | 0.2075 | 4.0480 | 2.2622 | 1.7858 |
| 1.4780 | 0.0720 | 0.1789 | 0.0147 | 0.0715 | 2.0022 | 0.9538 | 1.0483 |
100 | | 0.4754 | 0.0583 | 0.4828 | 0.0491 | 0.0577 | 0.9461 | 0.0047 | 0.9414 |
| 3.0700 | 0.1330 | 0.1216 | 0.0233 | 0.1281 | 3.7715 | 2.3684 | 1.4032 |
| 1.5032 | 0.0536 | 0.1543 | 0.0021 | 0.0536 | 1.9568 | 1.0495 | 0.9073 |
200 | | 0.4757 | 0.0277 | 0.3328 | 0.0485 | 0.0271 | 0.7984 | 0.1531 | 0.6453 |
| 3.0194 | 0.0743 | 0.0909 | 0.0065 | 0.0740 | 3.5524 | 2.4864 | 1.0660 |
| 1.5075 | 0.0326 | 0.1203 | 0.0050 | 0.0325 | 1.8609 | 1.1541 | 0.7069 |
500 | | 0.4786 | 0.0107 | 0.2073 | 0.0428 | 0.0103 | 0.6773 | 0.2799 | 0.3975 |
| 2.9967 | 0.0324 | 0.0600 | 0.0011 | 0.0324 | 3.3497 | 2.6438 | 0.7058 |
| 1.5114 | 0.0119 | 0.0729 | 0.0076 | 0.0118 | 1.7244 | 1.2984 | 0.4260 |
Table 6.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
Table 6.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different , and .
n | | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 0.1574 | 0.0120 | 0.7288 | 0.0495 | 0.0119 | 0.3712 | 0 | 0.3712 |
| 1.5322 | 0.0539 | 0.1548 | 0.0215 | 0.0529 | 1.9829 | 1.0815 | 0.9014 |
| 0.8033 | 0.0268 | 0.2048 | 0.0041 | 0.0268 | 1.1243 | 0.4822 | 0.6421 |
60 | | 0.1624 | 0.0110 | 0.7003 | 0.0827 | 0.0109 | 0.3669 | 0 | 0.3669 |
| 1.5095 | 0.0332 | 0.1214 | 0.0063 | 0.0331 | 1.8659 | 1.1531 | 0.7128 |
| 0.7839 | 0.0177 | 0.1662 | 0.0202 | 0.0174 | 1.0426 | 0.5252 | 0.5174 |
100 | | 0.1667 | 0.0089 | 0.6272 | 0.1116 | 0.0086 | 0.3482 | 0 | 0.3482 |
| 1.5018 | 0.0207 | 0.0960 | 0.0012 | 0.0207 | 1.7840 | 1.2196 | 0.5644 |
| 0.7834 | 0.0120 | 0.1366 | 0.0208 | 0.0117 | 0.9951 | 0.5716 | 0.4235 |
200 | | 0.1674 | 0.0054 | 0.4876 | 0.1161 | 0.0051 | 0.3066 | 0.0282 | 0.2785 |
| 1.5014 | 0.0123 | 0.0740 | 0.0010 | 0.0123 | 1.7189 | 1.2840 | 0.4349 |
| 0.7848 | 0.0067 | 0.1021 | 0.0191 | 0.0064 | 0.9421 | 0.6274 | 0.3147 |
500 | | 0.1594 | 0.0025 | 0.3328 | 0.0627 | 0.0024 | 0.2555 | 0.0633 | 0.1922 |
| 1.4980 | 0.0048 | 0.0462 | 0.0014 | 0.0048 | 1.6339 | 1.3621 | 0.2718 |
| 0.7953 | 0.0026 | 0.0635 | 0.0059 | 0.0026 | 0.8945 | 0.6961 | 0.1984 |
Table 7.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different , , and .
Table 7.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different , , and .
| Function | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 0.4664 | 0.0027 | 0.1092 | 0.0078 | 0.0025 | 0.5672 | 0.3655 | 0.2017 |
| 1.2808 | 0.0400 | 0.1597 | 0.0367 | 0.0379 | 1.6624 | 0.8992 | 0.7632 |
| 1.1119 | 0.0169 | 0.1825 | 0.0149 | 0.0167 | 1.3649 | 0.8590 | 0.5059 |
60 | | 0.4681 | 0.0016 | 0.0832 | 0.0042 | 0.0016 | 0.5472 | 0.3889 | 0.1583 |
| 1.2599 | 0.0233 | 0.1219 | 0.0198 | 0.0227 | 1.5551 | 0.9647 | 0.5905 |
| 1.1042 | 0.0106 | 0.1393 | 0.0079 | 0.0105 | 1.3050 | 0.9035 | 0.4015 |
100 | | 0.4678 | 0.0008 | 0.0734 | 0.0048 | 0.0008 | 0.5237 | 0.4118 | 0.1120 |
| 1.2506 | 0.0130 | 0.1063 | 0.0124 | 0.0127 | 1.4716 | 1.0296 | 0.4420 |
| 1.0971 | 0.0069 | 0.1039 | 0.0014 | 0.0069 | 1.2601 | 0.9342 | 0.3259 |
200 | | 0.4696 | 0.0006 | 0.0526 | 0.0010 | 0.0006 | 0.5169 | 0.4223 | 0.0946 |
| 1.2400 | 0.0080 | 0.0908 | 0.0038 | 0.0079 | 1.4147 | 1.0654 | 0.3493 |
| 1.0966 | 0.0044 | 0.0814 | 0.0009 | 0.0044 | 1.2273 | 0.9660 | 0.2613 |
500 | | 0.4690 | 0.0003 | 0.0513 | 0.0021 | 0.0003 | 0.4999 | 0.4382 | 0.0617 |
| 1.2383 | 0.0035 | 0.0722 | 0.0024 | 0.0035 | 1.3534 | 1.1232 | 0.2302 |
| 1.0934 | 0.0020 | 0.0537 | 0.0020 | 0.0020 | 1.1811 | 1.0057 | 0.1754 |
Table 8.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of Xg-BXII distribution for different , , and .
Table 8.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of Xg-BXII distribution for different , , and .
| Function | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 0.7728 | 0.0037 | 0.0761 | 0.0030 | 0.0037 | 0.8915 | 0.6540 | 0.2376 |
| 1.1087 | 0.0484 | 0.1888 | 0.0352 | 0.0468 | 1.5327 | 0.6846 | 0.8480 |
| 3.9365 | 0.8867 | 0.0556 | 0.0058 | 0.8862 | 5.7816 | 2.0914 | 3.6902 |
60 | | 0.7762 | 0.0022 | 0.0795 | 0.0015 | 0.0022 | 0.8690 | 0.6835 | 0.1856 |
| 1.1148 | 0.0323 | 0.1954 | 0.0299 | 0.0311 | 1.4605 | 0.7691 | 0.6913 |
| 3.9736 | 0.5880 | 0.0454 | 0.0036 | 0.5877 | 5.4762 | 2.4710 | 3.0052 |
100 | | 0.7746 | 0.0014 | 0.0611 | 0.0006 | 0.0014 | 0.8482 | 0.7011 | 0.1471 |
| 1.1383 | 0.0235 | 0.1564 | 0.0094 | 0.0234 | 1.4381 | 0.8386 | 0.5995 |
| 3.9758 | 0.3862 | 0.0387 | 0.0042 | 0.3860 | 5.1935 | 2.7582 | 2.4353 |
200 | | 0.7747 | 0.0007 | 0.0484 | 0.0004 | 0.0007 | 0.8265 | 0.7230 | 0.1035 |
| 1.1458 | 0.0154 | 0.1334 | 0.0029 | 0.0154 | 1.3887 | 0.9028 | 0.4859 |
| 3.9639 | 0.1742 | 0.0313 | 0.0012 | 0.1741 | 4.7818 | 3.1460 | 1.6358 |
500 | | 0.7754 | 0.0003 | 0.0341 | 0.0005 | 0.0003 | 0.8104 | 0.7404 | 0.0700 |
| 1.1506 | 0.0069 | 0.1079 | 0.0013 | 0.0069 | 1.3137 | 0.9876 | 0.3261 |
| 3.9813 | 0.0601 | 0.0210 | 0.0056 | 0.0596 | 4.4599 | 3.5027 | 0.9572 |
Table 9.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different , , and .
Table 9.
ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different , , and .
| Function | Average | ER | RE | RAB | Variance | UL | LL | Length |
---|
30 | | 0.7757 | 0.0024 | 0.0550 | 0.0023 | 0.0024 | 0.8721 | 0.6794 | 0.1927 |
| 0.6493 | 0.0126 | 0.4097 | 0.0058 | 0.0126 | 0.8689 | 0.4297 | 0.4392 |
| 2.3043 | 0.1900 | 0.0497 | 0.0217 | 0.1876 | 3.1532 | 1.4554 | 1.6979 |
60 | | 0.7779 | 0.0016 | 0.0622 | 0.0005 | 0.0016 | 0.8566 | 0.6991 | 0.1576 |
| 0.6362 | 0.0079 | 0.1696 | 0.0145 | 0.0078 | 0.8091 | 0.4633 | 0.3458 |
| 2.2673 | 0.1195 | 0.0393 | 0.0053 | 0.1193 | 2.9444 | 1.5903 | 1.3542 |
100 | | 0.7774 | 0.0011 | 0.0517 | 0.0001 | 0.0011 | 0.8419 | 0.7129 | 0.1290 |
| 0.6371 | 0.0051 | 0.1374 | 0.0131 | 0.0051 | 0.7765 | 0.4978 | 0.2788 |
| 2.2514 | 0.0754 | 0.0318 | 0.0018 | 0.0754 | 2.7897 | 1.7132 | 1.0765 |
200 | | 0.7781 | 0.0007 | 0.0423 | 0.0008 | 0.0007 | 0.8289 | 0.7273 | 0.1016 |
| 0.6381 | 0.0030 | 0.1109 | 0.0116 | 0.0029 | 0.7441 | 0.5321 | 0.2120 |
| 2.2535 | 0.0462 | 0.0242 | 0.0009 | 0.0462 | 2.6749 | 1.8320 | 0.8429 |
500 | | 0.7768 | 0.0003 | 0.0333 | 0.0009 | 0.0003 | 0.8081 | 0.7455 | 0.0626 |
| 0.6442 | 0.0011 | 0.0846 | 0.0021 | 0.0011 | 0.7095 | 0.5790 | 0.1305 |
| 2.2481 | 0.0182 | 0.0148 | 0.0032 | 0.0181 | 2.5119 | 1.9843 | 0.5276 |
Table 10.
Failure times of Wang’s data.
Table 10.
Failure times of Wang’s data.
Failure Times |
---|
5 | 11 | 21 | 31 | 46 | 75 | 98 | 122 | 145 |
165 | 196 | 224 | 245 | 293 | 321 | 330 | 350 | 420 |
Table 11.
Summary measures of Wang’s data.
Table 11.
Summary measures of Wang’s data.
Min | | | | | Max |
---|
5 | 53.25 | 155 | 172.11 | 281 | 420 |
Table 12.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Wang’s data.
Table 12.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Wang’s data.
Model | | MLE | SE | K-S | p-Value | AIC | BIC | CAIC |
---|
Xg-BXII | | 0.0134 | 0.0024 | 0.0980 | 0.9883 | 227.1598 | 229.8309 | 228.8741 |
| 0.9894 | 0.7263 |
| 0.0565 | 0.0511 |
LogL-W | | 0 | 3.6511 × 102 | 0.1132 | 0.9551 | 228.8923 | 232.4538 | 231.9691 |
| 0.8331 | 6.5613 × 101 |
| 0.0026 | 8.3356 × 10−4 |
| 1.1462 | 7.0112 × 10−2 |
L-W | | 0.0390 | 0.1127 | 0.1144 | 0.9512 | 228.6825 | 232.244 | 231.7594 |
| 0.0375 | 0.1227 |
| 0.0023 | 0.0008 |
| 1.1538 | 0.0690 |
AW | | 1.1424 | 0.2285 | 0.1177 | 0.9539 | 228.8920 | 232.4535 | 231.9689 |
| 0.0056 | 0.0012 |
| 3.8917 | 3.1294 × 102 |
| 0 | 1.7617 × 102 |
TXg | | 0.0168 | 0.0023 | 0.2292 | 0.2586 | 236.7577 | 238.5384 | 237.5577 |
| 1.6744 | 0.0159 |
Xg | | 0.0168 | 0.0023 | 0.2263 | 0.2720 | 233.3785 | 234.2689 | 233.6285 |
BXII | | 3.4236 | 7.0650 | 0.3679 | 0.0103 | 262.1132 | 263.8939 | 262.9132 |
| 0.0363 | 0.2104 |
Table 13.
Failure times of Meeker and Escober’s data.
Table 13.
Failure times of Meeker and Escober’s data.
Failure Times |
---|
0.1 | 0.2 | 1 | 2 | 3 | 6 | 7 | 11 | 12 | 18 |
21 | 32 | 36 | 40 | 45 | 46 | 47 | 50 | 55 | 60 |
63 | 67 | 72 | 75 | 79 | 82 | 83 | 84 | 85 | 86 |
Table 14.
Summary measures of Meeker and Escober’s data.
Table 14.
Summary measures of Meeker and Escober’s data.
Min | | | | | Max |
---|
0.1 | 11.25 | 45.50 | 42.28 | 70.75 | 86 |
Table 15.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Meeker and Escober’s data.
Table 15.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Meeker and Escober’s data.
Model | | MLE | SE | K-S | p-value | AIC | BIC | CAIC |
---|
Xg-BXII | | 0.0506 | 0.0073 | 0.1258 | 0.6859 | 279.5354 | 283.7389 | 280.4585 |
| 0.7605 | 0.3349 |
| 0.1183 | 0.0618 |
LogL-W | | 0.0721 | 0.0485 | 0.4869 | 0.0000 | 323.1017 | 328.7065 | 324.7017 |
| 0.2094 | 0.2475 |
| 0.0077 | 0.0068 |
| 1.2239 | 0.2054 |
L-W | | 0.0037 | 0.0169 | 0.1930 | 0.1875 | 292.5615 | 298.1663 | 294.1615 |
| 0.1799 | 0.2451 |
| 0.0283 | 0.0171 |
| 0.9544 | 0.1451 |
AW | | 1.6777 | 0.5348 | 0.1469 | 0.4912 | 282.8124 | 288.4172 | 284.4124 |
| 0.0158 | 0.0027 |
| 0.0028 | 0.0029 |
| 0.4609 | 0.1411 |
TXg | | 0.0515 | 0.0079 | 0.1446 | 0.5112 | 283.6904 | 286.4928 | 284.1348 |
| 0.0738 | 0.0467 |
Xg | | 0.0646 | 0.0071 | 0.2260 | 0.0793 | 296.3653 | 297.7665 | 296.5082 |
BXII | | 1.1138 | 0.3116 | 0.3026 | 0.0062 | 327.2089 | 330.0113 | 327.6533 |
| 0.2804 | 0.0905 |
Table 16.
Failure times of Murthy’s data.
Table 16.
Failure times of Murthy’s data.
Failure Times |
---|
0.03 | 0.12 | 0.22 | 0.35 | 0.73 | 0.79 | 1.25 | 1.41 | 1.25 | 1.79 |
1.80 | 1.94 | 2.38 | 2.40 | 2.87 | 2.99 | 3.14 | 3.19 | 4.72 | 5.09 |
Table 17.
Summary of measures of Murthy’s data.
Table 17.
Summary of measures of Murthy’s data.
Min | | | | | Max |
---|
0.03 | 0.775 | 1.795 | 1.923 | 2.900 | 5.090 |
Table 18.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Murthy’s data.
Table 18.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Murthy’s data.
Model | | MLE | SE | K-S | p-Value | AIC | BIC | CAIC |
---|
Xg-BXII | | 1.6585 | 0.4854 | 0.0793 | 0.9985 | 68.2122 | 71.1994 | 69.7122 |
| 1.4737 | 0.3825 |
| 0.7792 | 0.5698 |
LogL-W | | 0.7653 | 0.3680 | 0.0943 | 0.9866 | 70.6350 | 74.6179 | 73.3017 |
| 3.5675 | 4.4914 |
| 0.0618 | 0.0962 |
| 2.3888 | 0.9450 |
L-W | | 0.8038 | 0.9719 | 0.0928 | 0.9888 | 71.1862 | 75.1646 | 73.8483 |
| 0.4535 | 0.6129 |
| 0.0658 | 0.0906 |
| 2.2808 | 0.8595 |
AW | | 2.4874 | 1.1855 | 0.0949 | 0.9858 | 70.6935 | 74.6764 | 73.3602 |
| 0.2950 | 0.0954 |
| 0.2214 | 0.2859 |
| 0.7303 | 0.3590 |
TXg | | 1.1999 | 0.2288 | 0.0939 | 0.9873 | 69.4791 | 71.4706 | 70.1850 |
| 2.2659 | 2.0125 |
Xg | | 1.0484 | 0.1611 | 0.1447 | 0.7439 | 71.2099 | 72.2056 | 71.4321 |
BXII | | 1.4535 | 0.3065 | 0.2328 | 0.1952 | 77.0271 | 79.0185 | 77.7329 |
| 0.8763 | 0.2231 |
Table 19.
Drought mortality rates of the United Kingdom’s data.
Table 19.
Drought mortality rates of the United Kingdom’s data.
Drought Mortality Rates |
---|
0.0587 | 0.0863 | 0.1165 | 0.1247 | 0.1277 | 0.1303 | 0.1652 | 0.2079 | 0.2395 | 0.2751 | 0.2845 |
0.2992 | 0.3188 | 0.3317 | 0.3446 | 0.3553 | 0.3622 | 0.3926 | 0.3629 | 0.4110 | 0.4633 | 0.4690 |
0.4954 | 0.5139 | 0.5696 | 0.5837 | 0.6197 | 0.6365 | 0.7096 | 0.7193 | 0.7444 | 0.8590 | 1.0438 |
1.0602 | 1.1305 | 1.1468 | 1.1533 | 1.2260 | 1.2707 | 1.3423 | 1.4149 | 1.5709 | 1.6017 | 1.6083 |
1.6324 | 1.6998 | 1.8164 | 1.8392 | 1.8721 | 1.9844 | 2.1360 | 2.3987 | 2.4153 | 2.5225 | 2.7087 |
2.7946 | 3.3609 | 3.3715 | 3.7840 | 3.9042 | 4.1969 | 4.3451 | 4.4627 | 4.6477 | 5.3664 | 5.4500 |
5.7522 | 6.4241 | 7.0657 | 7.4456 | 8.2307 | 9.6315 | 10.1870 | 11.1429 | 11.2019 | 22.4584 | |
Table 20.
Summary measures of the United Kingdom’s data.
Table 20.
Summary measures of the United Kingdom’s data.
Min | | | | | Max |
---|
0.0587 | 0.4064 | 1.2484 | 2.4372 | 3.3636 | 11.4584 |
Table 21.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of the United Kingdom’s data.
Table 21.
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of the United Kingdom’s data.
Model | | MLE | SE | K-S | p-Value | AIC | BIC | AICc |
---|
Xg-BXII | | 0.3454 | 0.1043 | 0.0609 | 0.9242 | 283.9968 | 290.9890 | 284.3301 |
| 1.2184 | 0.1685 |
| 0.6902 | 0.1452 |
LogL-W | | 0.8673 | 0.2271 | 0.0800 | 0.6847 | 291.1339 | 300.4569 | 291.6973 |
| 9.6291 | 0.9726 |
| 0.3803 | 0.1367 |
| 0.8697 | 0.1023 |
L-W | | 2.5131 | 23.8014 | 0.0797 | 0.6702 | 291.4390 | 300.7620 | 292.0024 |
| 0 | 0.0602 |
| 0.5547 | 0.3752 |
| 0.8610 | 0.1268 |
AW | | 0.8466 | 0.1787 | 0.0806 | 0.6764 | 291.5037 | 300.8266 | 292.0671 |
| 0.1649 | 6.4435 |
| 0.2333 | 6.7957 |
| 0.8467 | 0.1421 |
TXg | | 0.5412 | 0.0890 | 0.1279 | 0.1523 | 288.5941 | 293.2556 | 288.7585 |
| 0.1028 | 0.0864 |
Xg | | 0.8154 | 0.0643 | 0.1904 | 0.0069 | 302.7674 | 305.0982 | 302.8215 |
BXII | | 1.3326 | 0.1420 | 0.0686 | 0.8431 | 288.9744 | 293.6359 | 289.1388 |
| 0.9017 | 0.1171 |