Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions
Abstract
:1. Introduction
2. A Cumulative Exposure Model of NH Distribution
- -
- Under usual conditions, the lifetime of a unit follows NH().
- -
- The progressive-stress (t) is directly proportional to the time t with constant rate , i.e., .
- -
- The scale parameter of the CDF in (2) satisfies the inverse power law, as follows
- -
- It is assumed that a and b are unknown physical positive parameters and need to be estimated.
- -
- Assume n is the total number of units tested, are the stress levels in the test, and is the use stress. Under each progressive-stress level, identical units are tested, and the progressive type II censoring is performed as follows: When the first failure occurs, units are picked at random from the remaining surviving units. When the second failure occurs, items from the remaining units are withdrawn at random. When the failure occurs, , the test is terminated, and all remaining items are removed.
- -
- The complete samples and type II censored samples are clearly specific examples of this technique. Under the progressive-stress , the observed progressive-censoring data are < , .
- -
- The linear cumulative exposure model (CEM) accounts for the effect of changing stress; for more details, see [13].
- -
- The CDF in progressive stress, , and the linear cumulative exposure model is given as followsThe corresponding PDF is given by
3. Maximum Likelihood Estimation
4. Bayesian Estimation
Bayes Estimation Using BSEL and BLINEX Loss Functions
- Step 1:
- For the parameters (, ), set the initial guess to (, ).
- Step 2:
- Set j = 1.
- Step 3:
- Create and where is the variance–covariance matrix.
- Step 4:
- Compute .
- Step 5:
- With probability accept (, ),
- Step 6:
- To obtain B number of samples for the parameters (, ), repeat steps (3) to (5) B times.
5. Interval Estimation
5.1. Asymptotic Confidence Interval
5.2. HPD Interval of Credibility
5.3. Bootstrap Confidence Intervals
- (1)
- Calculate the MLE values of the parameters using the original data , and .
- (2)
- To make a bootstrap sample use the variables , and .
- (3)
- The bootstrap estimates , , and , respectively, are obtained based on .
- (4)
- To obtain the bootstrap samples, repeat steps 1–3 several times and organize each estimate in ascending orderThe percentile bootstrap CIs for are then calculated as follows:
6. Simulation Study
- Step 1:
- Using the algorithm presented in [10], progressively type II censored random samples are generated from the uniform (0,1) distribution , for given values of .
- Step 2:
- To compare the performance of the estimation procedures developed in the study, we consider the following two schemes for each stress:Scheme 1: , .Scheme 2: , .
- Step 3:
- Progressively type II censored random samples are produced, and from inverse CDF (3), we specify the values of parameters as follows:In Table 1 .In Table 2 , andIn Table 3 , and .In Table 4 ( and .In Table 5 (), , and .In Table 6 (, , and .
- Step 4:
- The MLEs are obtained numerically by solving the likelihood equations with respect to () in (8)–(10) by using an iterative Newton–Raphson algorithm using the maxlik function of the “maxlik” package in the R program; for more information in this topic see [30].
- Step 5:
- Based on (15)–(17), and the MH algorithm, the Bayesian estimations with the BSEL and BLINEX loss functions of the parameters () are computed by (13) and (14), respectively.
- Step 6:
- The above steps are repeated I times based on I different samples, and then the average of likelihood and Bayesian estimations are computed, with their MSE, bias, and length of confidence intervals (LCI) of the parameters ().
- Step 7:
- In length of CI (LCI) of the MLE of each parameter, we compute the ACI for likelihood estimators and bootstrap CIs with the percentile algorithm and t algorithm, which can be denoted as LBP and LBT, respectively. In the LCI of Bayesian estimation, we compute the HPD for each loss function, denoted by the LCCI.
k = 2 | θ = 1.7, a = 1.3, b = 2 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15 | 1 | 65% | θ | −0.6751 | 1.5008 | −0.1938 | 0.0886 | −0.1172 | 0.0801 | −0.2619 | 0.1099 | 4.0094 | 0.1331 | 0.1271 | 0.8592 | 0.9821 | 0.7748 |
a | 0.3824 | 1.5335 | −0.0521 | 0.0788 | 0.0209 | 0.0994 | −0.1174 | 0.0740 | 4.6193 | 0.1432 | 0.1453 | 1.0048 | 1.1145 | 0.8922 | |||
b | 0.0673 | 0.6760 | −0.2972 | 0.1145 | −0.2616 | 0.0964 | −0.3314 | 0.1345 | 3.2139 | 0.0998 | 0.0993 | 0.6315 | 0.6551 | 0.6181 | |||
85% | θ | −0.4879 | 0.7308 | −0.0879 | 0.0657 | −0.0097 | 0.0736 | −0.1581 | 0.0724 | 2.7529 | 0.0899 | 0.0894 | 0.9552 | 1.0524 | 0.8647 | ||
a | 0.5008 | 0.9984 | 0.0504 | 0.0612 | 0.0195 | 0.0890 | 0.0426 | 0.0691 | 3.3911 | 0.1105 | 0.1101 | 0.9821 | 0.9834 | 0.9084 | |||
b | 0.0625 | 0.6469 | −0.1622 | 0.0558 | −0.1260 | 0.0471 | −0.1967 | 0.0667 | 3.1026 | 0.0978 | 0.0965 | 0.6718 | 0.6934 | 0.6551 | |||
2 | 65% | θ | 0.0872 | 0.5633 | 0.0550 | 0.0583 | 0.1391 | 0.0909 | −0.0206 | 0.0450 | 2.9235 | 0.0894 | 0.0875 | 0.9262 | 1.0470 | 0.8281 | |
a | −0.2725 | 0.5104 | 0.0517 | 0.1058 | 0.1262 | 0.1418 | −0.0158 | 0.0866 | 2.5902 | 0.0805 | 0.0818 | 1.2014 | 1.3056 | 1.0823 | |||
b | 0.5035 | 0.8594 | 0.0607 | 0.0411 | 0.1011 | 0.0500 | 0.0218 | 0.0360 | 3.0529 | 0.0908 | 0.0915 | 0.7305 | 0.7490 | 0.7048 | |||
85% | θ | 0.0823 | 0.5078 | 0.0542 | 0.0580 | 0.1346 | 0.0879 | −0.0171 | 0.0448 | 2.8345 | 0.0804 | 0.0803 | 0.9137 | 0.9103 | 0.8448 | ||
a | −0.1801 | 0.4902 | 0.0863 | 0.1041 | 0.1611 | 0.1509 | 0.0185 | 0.0778 | 2.6534 | 0.0858 | 0.0860 | 1.0977 | 1.2237 | 1.0492 | |||
b | 0.3257 | 0.5593 | 0.0494 | 0.0363 | 0.0879 | 0.0437 | 0.0126 | 0.0323 | 2.6404 | 0.0854 | 0.0845 | 0.7149 | 0.7384 | 0.6916 | |||
40, 50 | 1 | 65% | θ | −0.9678 | 1.4864 | −0.3139 | 0.1585 | −0.2489 | 0.1345 | −0.3726 | 0.1900 | 3.7765 | 0.1250 | 0.1174 | 0.9207 | 1.0006 | 0.8607 |
a | 0.3126 | 1.0050 | 0.1260 | 0.1244 | 0.1974 | 0.1754 | 0.0601 | 0.0913 | 4.4408 | 0.1433 | 0.1373 | 1.2116 | 1.3391 | 1.0992 | |||
b | −0.0262 | 0.5537 | −0.4534 | 0.2334 | −0.4262 | 0.2114 | −0.4791 | 0.2558 | 2.9167 | 0.0938 | 0.0929 | 0.6567 | 0.6774 | 0.6325 | |||
85% | θ | −0.9531 | 1.0410 | −0.2143 | 0.1193 | −0.1488 | 0.1099 | −0.2738 | 0.1380 | 1.4282 | 0.0451 | 0.0455 | 1.0419 | 1.1471 | 0.9624 | ||
a | 0.3033 | 0.9427 | 0.1230 | 0.1202 | 0.1374 | 0.1701 | 0.0582 | 0.0902 | 3.1609 | 0.1034 | 0.1027 | 1.2028 | 1.1908 | 1.1697 | |||
b | 0.2468 | 0.3993 | −0.2597 | 0.1017 | −0.2310 | 0.0904 | −0.2869 | 0.1145 | 2.2816 | 0.0720 | 0.0720 | 0.7346 | 0.7532 | 0.7076 | |||
2 | 65% | θ | 0.0189 | 0.5195 | 0.0350 | 0.0660 | 0.1067 | 0.0902 | −0.0307 | 0.0557 | 2.8259 | 0.0875 | 0.0878 | 0.9662 | 1.0601 | 0.9024 | |
a | −0.1043 | 0.3458 | 0.0534 | 0.1026 | 0.1154 | 0.1347 | −0.0034 | 0.0834 | 2.2699 | 0.0749 | 0.0750 | 1.1326 | 1.2150 | 1.0415 | |||
b | 0.2988 | 0.3316 | 0.0806 | 0.0416 | 0.1104 | 0.0489 | 0.0519 | 0.0365 | 1.9309 | 0.0610 | 0.0610 | 0.7083 | 0.7181 | 0.6939 | |||
85% | θ | 0.0199 | 0.2902 | 0.0345 | 0.0627 | 0.1023 | 0.0901 | −0.0155 | 0.0547 | 2.1112 | 0.0691 | 0.0689 | 0.9521 | 0.9602 | 0.9010 | ||
a | −0.0041 | 0.2644 | 0.0499 | 0.1013 | 0.1106 | 0.1325 | 0.0046 | 0.0829 | 2.0168 | 0.0606 | 0.0617 | 1.1228 | 1.1344 | 1.1043 | |||
b | 0.1356 | 0.1506 | 0.0437 | 0.0332 | 0.0711 | 0.0381 | 0.0173 | 0.0302 | 1.4262 | 0.0475 | 0.0476 | 0.6648 | 0.6811 | 0.6482 |
k = 2 | θ = 0.8, a = 0.5, b = 1.3 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15 | 1 | 65% | θ | −0.6718 | 2.8664 | −0.1568 | 0.1203 | −0.1274 | 0.1278 | −0.1845 | 0.1151 | 6.0949 | 0.2060 | 0.1973 | 0.9950 | 1.0839 | 0.9307 |
a | 0.2440 | 1.6298 | 0.1466 | 0.1261 | 0.1736 | 0.1496 | 0.1202 | 0.1050 | 4.9146 | 0.1549 | 0.1554 | 1.0924 | 1.1438 | 1.0436 | |||
b | 0.0957 | 0.4839 | −0.0738 | 0.0664 | −0.0268 | 0.0708 | −0.1174 | 0.0675 | 2.7025 | 0.0858 | 0.0848 | 0.9578 | 1.0205 | 0.8990 | |||
85% | θ | −0.2515 | 0.1789 | −0.0586 | 0.1017 | −0.0262 | 0.1189 | −0.0889 | 0.0893 | 1.3336 | 0.0416 | 0.0414 | 1.0767 | 1.1327 | 0.9832 | ||
a | 0.1999 | 0.2632 | 0.1647 | 0.1249 | 0.1903 | 0.1468 | 0.1398 | 0.1054 | 1.8532 | 0.0598 | 0.0607 | 1.1127 | 1.1571 | 1.0571 | |||
b | 0.2494 | 0.3102 | 0.0044 | 0.0617 | 0.0480 | 0.0721 | −0.0363 | 0.0566 | 1.9532 | 0.0594 | 0.0579 | 0.9574 | 1.0115 | 0.8972 | |||
2 | 65% | θ | 0.1997 | 0.2701 | 0.1153 | 0.1081 | 0.1526 | 0.1338 | 0.0801 | 0.0876 | 1.8819 | 0.0604 | 0.0602 | 1.0837 | 1.1369 | 0.9657 | |
a | −0.1490 | 0.1301 | 0.0586 | 0.0855 | 0.0816 | 0.0993 | 0.0365 | 0.0737 | 1.2884 | 0.0409 | 0.0399 | 1.0231 | 1.0804 | 0.9543 | |||
b | 0.2680 | 0.2961 | 0.0933 | 0.0665 | 0.1342 | 0.0823 | 0.0551 | 0.0557 | 1.8574 | 0.0593 | 0.0599 | 0.9271 | 0.9837 | 0.8748 | |||
85% | θ | 0.1673 | 0.2162 | 0.0845 | 0.0495 | 0.1121 | 0.0621 | 0.0586 | 0.0400 | 1.7014 | 0.0522 | 0.0516 | 0.7796 | 0.8472 | 0.7347 | ||
a | −0.0699 | 0.0976 | 0.0664 | 0.0581 | 0.0850 | 0.0669 | 0.0486 | 0.0508 | 1.1941 | 0.0379 | 0.0380 | 0.8233 | 0.8664 | 0.7746 | |||
b | 0.1521 | 0.2192 | 0.0372 | 0.0230 | 0.0627 | 0.0273 | 0.0130 | 0.0206 | 1.7367 | 0.0583 | 0.0588 | 0.5765 | 0.6031 | 0.5623 | |||
40, 50 | 1 | 65% | θ | −0.6809 | 3.2570 | −0.3695 | 0.1771 | −0.3528 | 0.1723 | −0.3849 | 0.1822 | 6.5550 | 0.2254 | 0.2022 | 0.6022 | 0.6613 | 0.5605 |
a | 0.6260 | 1.1068 | 0.3023 | 0.1956 | 0.3278 | 0.2238 | 0.2769 | 0.1692 | 3.3163 | 0.1154 | 0.1066 | 1.1525 | 1.2097 | 1.0901 | |||
b | 0.2043 | 0.1752 | 0.0435 | 0.0965 | 0.0945 | 0.1144 | −0.0056 | 0.0849 | 1.4326 | 0.0473 | 0.0469 | 1.1800 | 1.2411 | 1.1403 | |||
85% | θ | −0.4415 | 0.2093 | −0.2621 | 0.0908 | −0.2500 | 0.0879 | −0.2733 | 0.0942 | 0.4708 | 0.0153 | 0.0151 | 0.5155 | 0.5453 | 0.4948 | ||
a | 0.4369 | 0.3465 | 0.2701 | 0.1401 | 0.2903 | 0.1585 | 0.2502 | 0.1231 | 1.5470 | 0.0474 | 0.0479 | 0.9661 | 1.0058 | 0.9208 | |||
b | 0.2543 | 0.1467 | 0.0283 | 0.0340 | 0.0554 | 0.0390 | 0.0022 | 0.0311 | 1.1229 | 0.0349 | 0.0349 | 0.7075 | 0.7235 | 0.6965 | |||
2 | 65% | θ | 0.0872 | 0.1598 | 0.1037 | 0.0866 | 0.1337 | 0.1045 | 0.0753 | 0.0720 | 1.5300 | 0.0468 | 0.0472 | 0.9517 | 1.0225 | 0.9087 | |
a | −0.1012 | 0.0663 | 0.0297 | 0.0519 | 0.0452 | 0.0580 | 0.0149 | 0.0467 | 0.9282 | 0.0277 | 0.0276 | 0.8375 | 0.8677 | 0.7991 | |||
b | 0.1905 | 0.2101 | 0.0796 | 0.0639 | 0.1103 | 0.0757 | 0.0506 | 0.0552 | 1.6350 | 0.0535 | 0.0526 | 0.8925 | 0.9435 | 0.8638 | |||
85% | θ | 0.0839 | 0.1248 | 0.0757 | 0.0492 | 0.0973 | 0.0593 | 0.0550 | 0.0409 | 1.3462 | 0.0423 | 0.0426 | 0.7645 | 0.8162 | 0.7216 | ||
a | −0.0484 | 0.0473 | 0.0579 | 0.0489 | 0.0717 | 0.0550 | 0.0446 | 0.0435 | 0.8311 | 0.0265 | 0.0264 | 0.7049 | 0.7358 | 0.6820 | |||
b | 0.0961 | 0.1294 | 0.0243 | 0.0259 | 0.0433 | 0.0286 | 0.0059 | 0.0241 | 1.3594 | 0.0432 | 0.0432 | 0.6307 | 0.6517 | 0.6262 |
k = 2 | θ = 3, a = 0.5, b = 0.6 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15 | 1 | 65% | θ | −1.0081 | 1.4542 | −0.3499 | 0.1451 | −0.2693 | 0.0983 | −0.4242 | 0.2008 | 2.5956 | 0.0851 | 0.0847 | 0.5826 | 0.6182 | 0.5601 |
a | 0.0679 | 1.2060 | −0.3724 | 0.1673 | −0.2938 | 0.1205 | −0.4438 | 0.2221 | 4.2987 | 0.1386 | 0.1410 | 0.6487 | 0.7208 | 0.6095 | |||
b | −0.1176 | 0.1318 | −0.2613 | 0.0778 | −0.2521 | 0.0737 | −0.2702 | 0.0819 | 1.3473 | 0.0437 | 0.0433 | 0.3603 | 0.3730 | 0.3459 | |||
85% | θ | −0.4454 | 0.4194 | −0.1202 | 0.0252 | −0.0733 | 0.0168 | −0.1652 | 0.0375 | 1.8440 | 0.0591 | 0.0591 | 0.3991 | 0.4098 | 0.3920 | ||
a | −0.0649 | 0.2608 | −0.1168 | 0.0311 | −0.0670 | 0.0235 | −0.1639 | 0.0433 | 1.9865 | 0.0654 | 0.0637 | 0.5281 | 0.5430 | 0.5133 | |||
b | −0.0581 | 0.0912 | −0.1632 | 0.0352 | −0.1555 | 0.0331 | −0.1707 | 0.0374 | 1.1620 | 0.0367 | 0.0374 | 0.3568 | 0.3624 | 0.3499 | |||
2 | 65% | θ | −0.1533 | 0.4265 | −0.0128 | 0.0410 | 0.0808 | 0.0511 | −0.0995 | 0.0484 | 2.4897 | 0.0734 | 0.0726 | 0.8016 | 0.8379 | 0.7820 | |
a | −0.2400 | 0.6128 | −0.0100 | 0.0597 | 0.0877 | 0.0755 | −0.0987 | 0.0639 | 2.9224 | 0.0901 | 0.0898 | 0.9570 | 1.0232 | 0.9205 | |||
b | 0.2240 | 0.1830 | 0.0290 | 0.0270 | 0.0438 | 0.0293 | 0.0145 | 0.0253 | 1.4291 | 0.0427 | 0.0426 | 0.6181 | 0.6302 | 0.6014 | |||
85% | θ | −0.0076 | 0.2889 | 0.0133 | 0.0152 | 0.0632 | 0.0198 | −0.0346 | 0.0158 | 2.1077 | 0.0662 | 0.0651 | 0.4780 | 0.4812 | 0.4698 | ||
a | −0.1365 | 0.3228 | 0.0253 | 0.0269 | 0.0803 | 0.0347 | −0.0268 | 0.0255 | 2.1629 | 0.0679 | 0.0684 | 0.6379 | 0.6699 | 0.6282 | |||
b | 0.1288 | 0.1135 | 0.0058 | 0.0144 | 0.0155 | 0.0150 | −0.0039 | 0.0140 | 1.2206 | 0.0386 | 0.0384 | 0.4462 | 0.4534 | 0.4430 | |||
40, 50 | 1 | 65% | θ | −1.1973 | 2.0957 | −0.4388 | 0.2123 | −0.3633 | 0.1551 | −0.5082 | 0.2761 | 3.1917 | 0.1021 | 0.0996 | 0.5524 | 0.5850 | 0.5390 |
a | 0.4242 | 1.8779 | −0.4240 | 0.2012 | −0.3566 | 0.1528 | −0.4852 | 0.2543 | 5.1106 | 0.1639 | 0.1643 | 0.5646 | 0.6197 | 0.5296 | |||
b | −0.3040 | 0.1460 | −0.3420 | 0.1222 | −0.3367 | 0.1188 | −0.3471 | 0.1255 | 0.9077 | 0.0310 | 0.0305 | 0.2663 | 0.2715 | 0.2631 | |||
85% | θ | −1.1130 | 1.6602 | −0.1645 | 0.0369 | −0.1219 | 0.0252 | −0.2053 | 0.0517 | 2.5459 | 0.0791 | 0.0792 | 0.3912 | 0.3950 | 0.3890 | ||
a | 0.3356 | 0.3522 | −0.1170 | 0.0290 | −0.0745 | 0.0218 | −0.1575 | 0.0393 | 4.9114 | 0.1509 | 0.1504 | 0.4752 | 0.4874 | 0.4623 | |||
b | −0.1376 | 0.0633 | −0.2416 | 0.0645 | −0.2372 | 0.0625 | −0.2459 | 0.0665 | 0.8264 | 0.0268 | 0.0265 | 0.2928 | 0.2964 | 0.2898 | |||
2 | 65% | θ | −0.0736 | 0.3191 | −0.0093 | 0.0421 | 0.0786 | 0.0525 | −0.0899 | 0.0477 | 2.1964 | 0.0688 | 0.0681 | 0.8118 | 0.8396 | 0.7883 | |
a | −0.0731 | 0.2996 | 0.0195 | 0.0535 | 0.1048 | 0.0712 | −0.0585 | 0.0517 | 2.1275 | 0.0674 | 0.0659 | 0.8891 | 0.9502 | 0.8511 | |||
b | 0.1074 | 0.0593 | 0.0351 | 0.0180 | 0.0438 | 0.0190 | 0.0264 | 0.0171 | 0.8576 | 0.0288 | 0.0285 | 0.4791 | 0.4815 | 0.4721 | |||
85% | θ | −0.0128 | 0.2589 | 0.0097 | 0.0167 | 0.0557 | 0.0204 | −0.0344 | 0.0173 | 1.3010 | 0.0590 | 0.0589 | 0.5014 | 0.5122 | 0.4937 | ||
a | 0.0694 | 0.2707 | 0.0260 | 0.0300 | 0.0731 | 0.0363 | −0.0188 | 0.0284 | 1.3429 | 0.0612 | 0.0612 | 0.6600 | 0.6807 | 0.6407 | |||
b | 0.0647 | 0.0409 | 0.0031 | 0.0099 | 0.0090 | 0.0101 | −0.0027 | 0.0097 | 0.7513 | 0.0240 | 0.0241 | 0.3942 | 0.3955 | 0.3921 |
k = 4 | θ = 1.7, a = 1.3, b = 2 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2, n3, n4 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15, 18, 10 | 1 | 65% | θ | −0.8715 | 1.0614 | −0.2242 | 0.1068 | −0.1519 | 0.0943 | −0.2893 | 0.1302 | 2.1561 | 0.0965 | 0.0965 | 0.8956 | 0.9632 | 0.8047 |
a | 0.8096 | 1.4846 | 0.0001 | 0.0882 | 0.0688 | 0.1142 | −0.0625 | 0.0765 | 3.7822 | 0.1691 | 0.1694 | 1.1078 | 1.2537 | 0.9950 | |||
b | 0.1157 | 0.3767 | −0.3089 | 0.1190 | −0.2801 | 0.1037 | −0.3365 | 0.1355 | 2.3651 | 0.1070 | 0.1066 | 0.6109 | 0.6291 | 0.5869 | |||
85% | θ | −0.5590 | 0.7077 | −0.1089 | 0.0307 | −0.0677 | 0.0259 | −0.1476 | 0.0388 | 2.4667 | 0.1144 | 0.1135 | 0.5240 | 0.5597 | 0.4972 | ||
a | 0.6354 | 0.9014 | 0.0470 | 0.0402 | 0.0905 | 0.0539 | 0.0062 | 0.0323 | 3.0655 | 0.1441 | 0.1385 | 0.6795 | 0.7253 | 0.6408 | |||
b | 0.0962 | 0.3299 | −0.1872 | 0.0476 | −0.1704 | 0.0419 | −0.2036 | 0.0538 | 2.2221 | 0.1003 | 0.1017 | 0.4437 | 0.4436 | 0.4419 | |||
2 | 65% | θ | 0.1463 | 0.4847 | 0.0373 | 0.0756 | 0.1170 | 0.1094 | −0.0345 | 0.0617 | 2.9452 | 0.1315 | 0.1295 | 0.9935 | 1.0840 | 0.8752 | |
a | −0.3137 | 0.3755 | 0.0228 | 0.1139 | 0.0899 | 0.1463 | −0.0388 | 0.0959 | 2.0658 | 0.0946 | 0.0937 | 1.2244 | 1.3813 | 1.1134 | |||
b | 0.3591 | 0.4138 | 0.0913 | 0.0437 | 0.1225 | 0.0519 | 0.0613 | 0.0379 | 2.0944 | 0.0977 | 0.0973 | 0.7276 | 0.7407 | 0.7164 | |||
85% | θ | 0.1331 | 0.3552 | 0.0306 | 0.0226 | 0.0738 | 0.0306 | −0.0100 | 0.0192 | 2.8690 | 0.1216 | 0.1226 | 0.5557 | 0.6039 | 0.5304 | ||
a | −0.1476 | 0.3143 | 0.0460 | 0.0416 | 0.0882 | 0.0533 | 0.0067 | 0.0348 | 2.1222 | 0.1006 | 0.0992 | 0.7528 | 0.8065 | 0.7120 | |||
b | 0.2191 | 0.2586 | 0.0238 | 0.0139 | 0.0419 | 0.0154 | 0.0060 | 0.0132 | 1.8008 | 0.0761 | 0.0764 | 0.4427 | 0.4514 | 0.4406 | |||
25, 20, 20, 25 | 1 | 65% | θ | −1.0371 | 1.2431 | −0.2741 | 0.1331 | −0.2060 | 0.1158 | −0.3356 | 0.1606 | 1.6059 | 0.0765 | 0.0765 | 0.9131 | 1.0396 | 0.8033 |
a | 0.9794 | 0.9451 | 0.0684 | 0.1174 | 0.1367 | 0.1581 | 0.0052 | 0.0917 | 2.7529 | 0.1255 | 0.1214 | 1.1129 | 1.2247 | 1.0286 | |||
b | 0.2007 | 0.4168 | −0.3360 | 0.1360 | −0.3096 | 0.1204 | −0.3612 | 0.1525 | 2.4078 | 0.0999 | 0.1001 | 0.5456 | 0.5477 | 0.5355 | |||
85% | θ | −0.8327 | 0.8583 | −0.1666 | 0.0526 | −0.1276 | 0.0443 | −0.2031 | 0.0638 | 1.5933 | 0.0719 | 0.0724 | 0.6202 | 0.6524 | 0.5959 | ||
a | 0.8746 | 0.8324 | 0.0859 | 0.0508 | 0.1282 | 0.0662 | 0.0464 | 0.0406 | 2.9330 | 0.1322 | 0.1275 | 0.7932 | 0.8515 | 0.7411 | |||
b | 0.2137 | 0.3082 | −0.1874 | 0.0491 | −0.1719 | 0.0440 | −0.2027 | 0.0547 | 2.0106 | 0.0873 | 0.0873 | 0.4630 | 0.4716 | 0.4563 | |||
2 | 65% | θ | 0.0105 | 0.4456 | 0.0441 | 0.0754 | 0.1172 | 0.1023 | −0.0232 | 0.0628 | 2.6189 | 0.1169 | 0.1171 | 1.0020 | 1.0986 | 0.9428 | |
a | −0.1869 | 0.3433 | −0.0015 | 0.1093 | 0.0559 | 0.1339 | −0.0545 | 0.0955 | 2.1788 | 0.0930 | 0.0930 | 1.1695 | 1.2602 | 1.0874 | |||
b | 0.3025 | 0.2693 | 0.0981 | 0.0430 | 0.1247 | 0.0504 | 0.0725 | 0.0374 | 1.6543 | 0.0717 | 0.0713 | 0.6976 | 0.7207 | 0.6846 | |||
85% | θ | 0.0224 | 0.2149 | 0.0203 | 0.0238 | 0.0630 | 0.0310 | −0.0196 | 0.0212 | 1.8168 | 0.0821 | 0.0822 | 0.6106 | 0.6641 | 0.5580 | ||
a | −0.0716 | 0.2182 | 0.0536 | 0.0487 | 0.0929 | 0.0604 | 0.0168 | 0.0415 | 1.8115 | 0.0809 | 0.0810 | 0.7661 | 0.8193 | 0.7406 | |||
b | 0.1356 | 0.1320 | 0.0284 | 0.0175 | 0.0444 | 0.0191 | 0.0127 | 0.0164 | 1.3223 | 0.0592 | 0.0601 | 0.4837 | 0.4910 | 0.4805 |
k = 4 | θ = 0.8, a = 0.5, b = 1.3 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2, n3, n4 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15, 18, 10 | 1 | 65% | θ | −0.4626 | 0.2492 | −0.2234 | 0.1145 | −0.1991 | 0.1159 | −0.2461 | 0.1151 | 0.7363 | 0.0344 | 0.0328 | 0.8227 | 0.8569 | 0.7749 |
a | 0.3913 | 0.3915 | 0.2019 | 0.1463 | 0.2270 | 0.1703 | 0.1772 | 0.1240 | 1.9158 | 0.0833 | 0.0839 | 1.1213 | 1.1835 | 1.0524 | |||
b | 0.1880 | 0.1968 | −0.0547 | 0.0723 | −0.0107 | 0.0790 | −0.0959 | 0.0703 | 1.5765 | 0.0739 | 0.0737 | 0.9501 | 1.0167 | 0.8885 | |||
85% | θ | −0.2908 | 0.1596 | −0.1193 | 0.0548 | −0.1007 | 0.0563 | −0.1370 | 0.0545 | 1.0747 | 0.0506 | 0.0498 | 0.7117 | 0.7482 | 0.6735 | ||
a | 0.2322 | 0.2134 | 0.1603 | 0.0924 | 0.1795 | 0.1058 | 0.1415 | 0.0804 | 1.5669 | 0.0688 | 0.0679 | 0.8484 | 0.8828 | 0.8237 | |||
b | 0.1623 | 0.1711 | −0.0366 | 0.0257 | −0.0133 | 0.0263 | −0.0588 | 0.0263 | 1.4928 | 0.0685 | 0.0673 | 0.5739 | 0.5880 | 0.5635 | |||
2 | 65% | θ | 0.1299 | 0.1748 | 0.1374 | 0.1253 | 0.1716 | 0.1523 | 0.1044 | 0.1022 | 1.5594 | 0.0733 | 0.0726 | 1.0883 | 1.1717 | 1.0231 | |
a | −0.1473 | 0.0858 | 0.0164 | 0.0596 | 0.0339 | 0.0670 | −0.0005 | 0.0536 | 0.9932 | 0.0429 | 0.0434 | 0.8881 | 0.9247 | 0.8439 | |||
b | 0.2240 | 0.2153 | 0.0817 | 0.0631 | 0.1126 | 0.0743 | 0.0525 | 0.0551 | 1.5945 | 0.0698 | 0.0698 | 0.9148 | 0.9542 | 0.8811 | |||
85% | θ | 0.1153 | 0.1522 | 0.0809 | 0.0505 | 0.1036 | 0.0606 | 0.0592 | 0.0421 | 1.4724 | 0.0720 | 0.0710 | 0.7057 | 0.7504 | 0.6762 | ||
a | −0.0763 | 0.0775 | 0.0679 | 0.0594 | 0.0842 | 0.0672 | 0.0520 | 0.0526 | 1.0509 | 0.0466 | 0.0465 | 0.8191 | 0.8463 | 0.7834 | |||
b | 0.1143 | 0.1520 | 0.0234 | 0.0241 | 0.0429 | 0.0264 | 0.0046 | 0.0228 | 1.4627 | 0.0616 | 0.0614 | 0.5870 | 0.5948 | 0.5762 | |||
25, 20, 20, 25 | 1 | 65% | θ | −0.5191 | 0.2829 | −0.2937 | 0.1456 | −0.2745 | 0.1430 | −0.3117 | 0.1490 | 0.4551 | 0.0212 | 0.0194 | 0.8007 | 0.8642 | 0.7501 |
a | 0.5151 | 0.4880 | 0.2251 | 0.1420 | 0.2473 | 0.1616 | 0.2032 | 0.1237 | 1.8517 | 0.0845 | 0.0843 | 1.0437 | 1.0777 | 0.9973 | |||
b | 0.1694 | 0.1096 | −0.0074 | 0.0847 | 0.0334 | 0.0947 | −0.0465 | 0.0789 | 1.1161 | 0.0488 | 0.0475 | 1.1251 | 1.1954 | 1.0986 | |||
85% | θ | −0.3831 | 0.1711 | −0.2176 | 0.0699 | −0.2051 | 0.0671 | −0.2295 | 0.0729 | 0.6114 | 0.0285 | 0.0277 | 0.5299 | 0.5513 | 0.5081 | ||
a | 0.3178 | 0.2274 | 0.2112 | 0.1011 | 0.2299 | 0.1151 | 0.1928 | 0.0883 | 1.3947 | 0.0643 | 0.0637 | 0.8576 | 0.8927 | 0.8125 | |||
b | 0.1490 | 0.1018 | 0.0134 | 0.0287 | 0.0367 | 0.0318 | −0.0089 | 0.0269 | 1.0124 | 0.0450 | 0.0420 | 0.6588 | 0.6696 | 0.6415 | |||
2 | 65% | θ | 0.1017 | 0.1693 | 0.1241 | 0.0918 | 0.1547 | 0.1115 | 0.0951 | 0.0755 | 1.5643 | 0.0706 | 0.0690 | 1.0020 | 1.0791 | 0.9475 | |
a | −0.1549 | 0.0652 | −0.0193 | 0.0454 | −0.0055 | 0.0490 | −0.0326 | 0.0426 | 0.7969 | 0.0354 | 0.0353 | 0.7545 | 0.7783 | 0.7351 | |||
b | 0.2107 | 0.1870 | 0.0883 | 0.0566 | 0.1150 | 0.0655 | 0.0629 | 0.0499 | 1.4821 | 0.0700 | 0.0698 | 0.8434 | 0.8594 | 0.8319 | |||
85% | θ | 0.0992 | 0.1270 | 0.0802 | 0.0429 | 0.0992 | 0.0500 | 0.0619 | 0.0369 | 1.3428 | 0.0595 | 0.0595 | 0.7124 | 0.7547 | 0.6790 | ||
a | −0.0549 | 0.0520 | 0.0444 | 0.0417 | 0.0577 | 0.0465 | 0.0314 | 0.0374 | 0.8686 | 0.0394 | 0.0392 | 0.6876 | 0.7238 | 0.6680 | |||
b | 0.0887 | 0.1077 | 0.0285 | 0.0241 | 0.0455 | 0.0266 | 0.0120 | 0.0225 | 1.2399 | 0.0551 | 0.0552 | 0.5933 | 0.6070 | 0.5709 |
k = 4 | θ = 3, a = 2.5, b = 0.6 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n1, n2, n3, n4 | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |
20, 15, 18, 10 | 1 | 65% | θ | −1.3150 | 2.3042 | −0.3331 | 0.1341 | −0.2518 | 0.0897 | −0.4078 | 0.1876 | 2.9740 | 0.0931 | 0.0934 | 0.5858 | 0.6235 | 0.5653 |
a | 0.8989 | 3.0227 | −0.3491 | 0.1514 | −0.2737 | 0.1096 | −0.4177 | 0.2009 | 5.8365 | 0.1831 | 0.1819 | 0.6546 | 0.7089 | 0.6228 | |||
b | −0.0913 | 0.0606 | −0.2622 | 0.0764 | −0.2552 | 0.0731 | −0.2691 | 0.0798 | 0.8971 | 0.0275 | 0.0271 | 0.3234 | 0.3309 | 0.3182 | |||
85% | θ | −0.4708 | 0.4836 | −0.1116 | 0.0243 | −0.0654 | 0.0166 | −0.1557 | 0.0358 | 2.0072 | 0.0672 | 0.0661 | 0.4192 | 0.4311 | 0.4174 | ||
a | 0.1134 | 0.3691 | −0.0982 | 0.0274 | −0.0510 | 0.0217 | −0.1430 | 0.0372 | 2.3408 | 0.0769 | 0.0765 | 0.5218 | 0.5481 | 0.5058 | |||
b | −0.1124 | 0.0513 | −0.1800 | 0.0387 | −0.1749 | 0.0370 | −0.1852 | 0.0405 | 0.7709 | 0.0247 | 0.0247 | 0.2934 | 0.2941 | 0.2908 | |||
2 | 65% | θ | −0.1024 | 0.4661 | −0.0198 | 0.0413 | 0.0717 | 0.0510 | −0.1043 | 0.0486 | 2.6473 | 0.0861 | 0.0860 | 0.8006 | 0.8313 | 0.7613 | |
a | −0.1680 | 0.7442 | −0.0116 | 0.0668 | 0.0803 | 0.0824 | −0.0956 | 0.0693 | 3.3186 | 0.1090 | 0.1096 | 0.9976 | 1.0673 | 0.9505 | |||
b | 0.1489 | 0.0759 | 0.0331 | 0.0167 | 0.0426 | 0.0178 | 0.0237 | 0.0159 | 0.9086 | 0.0284 | 0.0285 | 0.4847 | 0.4906 | 0.4771 | |||
85% | θ | 0.0948 | 0.4053 | 0.0163 | 0.0164 | 0.0650 | 0.0211 | −0.0303 | 0.0166 | 2.4820 | 0.0808 | 0.0806 | 0.4926 | 0.5031 | 0.4903 | ||
a | −0.1387 | 0.3914 | 0.0311 | 0.0291 | 0.0823 | 0.0368 | −0.0174 | 0.0271 | 2.3925 | 0.0765 | 0.0772 | 0.6587 | 0.6836 | 0.6349 | |||
b | 0.0754 | 0.0501 | 0.0002 | 0.0083 | 0.0061 | 0.0085 | −0.0057 | 0.0082 | 0.8269 | 0.0263 | 0.0273 | 0.3591 | 0.3616 | 0.3564 | |||
25, 20, 20, 25 | 1 | 65% | θ | −1.0913 | 1.4984 | −0.3755 | 0.1641 | −0.2985 | 0.1155 | −0.4466 | 0.2206 | 2.1751 | 0.0671 | 0.0668 | 0.5890 | 0.6191 | 0.5681 |
a | 0.1753 | 0.6860 | −0.3636 | 0.1586 | −0.2929 | 0.1171 | −0.4282 | 0.2069 | 3.1748 | 0.1003 | 0.1006 | 0.6294 | 0.6672 | 0.6001 | |||
b | −0.1634 | 0.0712 | −0.2865 | 0.0879 | −0.2809 | 0.0849 | −0.2920 | 0.0909 | 0.8272 | 0.0254 | 0.0253 | 0.2929 | 0.2975 | 0.2878 | |||
85% | θ | −0.6934 | 0.7946 | −0.1339 | 0.0307 | −0.0891 | 0.0214 | −0.1768 | 0.0435 | 2.1972 | 0.0689 | 0.0680 | 0.4372 | 0.4488 | 0.4279 | ||
a | 0.1442 | 0.6040 | −0.0950 | 0.0278 | −0.0492 | 0.0224 | −0.1385 | 0.0372 | 3.0604 | 0.1013 | 0.0911 | 0.5472 | 0.5677 | 0.5313 | |||
b | −0.1170 | 0.0424 | −0.1894 | 0.0414 | −0.1851 | 0.0398 | −0.1936 | 0.0429 | 0.6642 | 0.0197 | 0.0199 | 0.2797 | 0.2791 | 0.2786 | |||
2 | 65% | θ | −0.1862 | 0.6036 | −0.0300 | 0.0419 | 0.0594 | 0.0488 | −0.1123 | 0.0510 | 2.9582 | 0.0956 | 0.0961 | 0.7994 | 0.8209 | 0.7782 | |
a | −0.0428 | 0.6685 | −0.0307 | 0.0599 | 0.0549 | 0.0696 | −0.1088 | 0.0656 | 3.2022 | 0.1008 | 0.1016 | 0.9397 | 1.0007 | 0.9127 | |||
b | 0.1438 | 0.0634 | 0.0441 | 0.0147 | 0.0515 | 0.0157 | 0.0367 | 0.0139 | 0.8105 | 0.0261 | 0.0261 | 0.4317 | 0.4365 | 0.4286 | |||
85% | θ | −0.0443 | 0.1186 | 0.0114 | 0.0177 | 0.0587 | 0.0219 | −0.0340 | 0.0182 | 1.3394 | 0.0427 | 0.0432 | 0.5242 | 0.5270 | 0.5207 | ||
a | −0.0603 | 0.1282 | 0.0288 | 0.0309 | 0.0779 | 0.0382 | −0.0179 | 0.0290 | 1.3843 | 0.0435 | 0.0431 | 0.6507 | 0.6920 | 0.6365 | |||
b | 0.0485 | 0.0237 | 0.0034 | 0.0071 | 0.0082 | 0.0072 | −0.0014 | 0.0070 | 0.5732 | 0.0183 | 0.0181 | 0.3187 | 0.3206 | 0.3177 |
Simulation Results
- (1)
- For fixed values of the sample sizes , by increasing the censored sample sizes, , the bias, MSE, and LCI of the estimates decrease for the two different censored schemes.
- (2)
- For fixed values of , by increasing the sample sizes , the bias, MSE, and LCI decrease for different censored schemes.
- (3)
- For fixed values of or or scheme, by increasing the level of stress k, the bias, MSE, and LCI decrease.
- (4)
- For fixed values of or or scheme, we note that Scheme 2 is better than Scheme 1 for some or all parameters.
- (5)
- The bias and MSE reduce significantly, and the symmetric and asymmetric Bayesian estimations are better than the MLE in the considered scenarios.
- (6)
- The LCI reduces significantly, the symmetric and asymmetric Bayesian estimations of the HPD are better than the ACI of MLE.
- (7)
- We observe that the shortest lengths of the CI are the bootstrap CI.
7. An Illustrative Example
8. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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First Level | Second Level | |||
---|---|---|---|---|
Estimates | SE | Estimates | SE | |
3.3823 | 0.8429 | 2.6078 | 0.5590 | |
a | ||||
b | 1.3756 | 0.0320 | 1.6994 | 0.0507 |
AIC | 554.7177 | 493.5818 | ||
KSD | 0.1124 | 0.1113 | ||
p-value | 0.3852 | 0.4362 |
MLE | Bayesian | |||||||
---|---|---|---|---|---|---|---|---|
Estimates | SE | Lower | Upper | Estimates | SE | Lower | Upper | |
θ | 11.9739 | 0.5590 | 0.0000 | 66.3388 | 12.3834 | 0.4691 | 4.0420 | 21.6353 |
a | ||||||||
b | 1.0260 | 0.0507 | 0.9979 | 1.0541 | 1.0322 | 0.0409 | 0.9997 | 1.0539 |
AIC | 1074.528 |
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Alotaibi, R.; Alamri, F.S.; Almetwally, E.M.; Wang, M.; Rezk, H. Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions. Mathematics 2022, 10, 1602. https://doi.org/10.3390/math10091602
Alotaibi R, Alamri FS, Almetwally EM, Wang M, Rezk H. Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions. Mathematics. 2022; 10(9):1602. https://doi.org/10.3390/math10091602
Chicago/Turabian StyleAlotaibi, Refah, Faten S. Alamri, Ehab M. Almetwally, Min Wang, and Hoda Rezk. 2022. "Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions" Mathematics 10, no. 9: 1602. https://doi.org/10.3390/math10091602