Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function
Abstract
:1. Introduction
2. Model Description and Generalized Type-I Progressive Hybrid Censoring
- Case A: If , then , , is deduced as followsThe sufficient condition for the function to be an accelerating function is (see Mann et al. [30])
- Case B: If , then, as shown in Case A, , , can be written in the formThe sufficient condition for the function to be an accelerating function is
- Case C: If , then, as shown in Case A, , , can be written in the formThe sufficient condition for the function to be an accelerating function is
2.1. Generalized Type-I Progressive Hybrid Censoring
- Suppose that a set n of randomly chosen units is subjected to a lifetime testing experiment.
- The units are initially tested at normal stress conditions up to a fixed time . If they do not fail by this time, then they are tested under accelerated conditions.
- Suppose that the experimental time and the integers and () are assigned before beginning the experiment, such that , .
- At the first failure time , operating units are selected at random and excluded from the experiment. At the next failure time , operating units are selected at random and excluded from the experiment, and this technique continues. Finally, the experiment is finished at the time , and all of the remaining operating units are excluded from the experiment. The values of the final censoring number are given in Table 1.
- Let r() be the number of units that fail under normal (accelerated) stress conditions before (after) time , and let D be the number of units that fail before time . Then, the experimental end time is given byOne of the next six cases may be noticed with the following observations:
- Case 1: If the experimental time is reached before the k-th failure time occurs and failures occur up to time , , then we will not eliminate any operating units from the experiment directly following the -th, ..., -th failure times and will eliminate all of the remaining operating units from the experiment at the k-th failure time, thereby stopping the experiment at , where ; see Figure 2. In this case, we permit the experiment to continue after experimental time is reached to ensure that at least the k-th failure time occurs. In this case, and the following observations will be noticed:
- Case 2: If the k-th failure time occurs between times and , , and failures occur up to time , then all of the remaining operating units = are eliminated from the experiment, thereby stopping the experiment at ; see Figure 3. In this case, and the following observations will be noticed:
- Case 3: If the k-th failure time occurs before time , , and failures occur up to time , then all of the remaining operating units = are eliminated from the experiment, thereby stopping the experiment at ; see Figure 4. In this case, and the following observations will be noticed:
- Case 4: If the k-th failure time occurs after time and the m-th failure time occurs before time , , then all of the remaining operating units =n−m− are eliminated from the experiment, thereby stopping the experiment at ; see Figure 5. In this case, and the following observations will be noticed:
- Case 5: If the k-th failure time occurs before time and the m-th failure time occurs between times and , , then all of the remaining operating units =n−m− are eliminated from the experiment, thereby stopping the experiment at ; see Figure 6. In this case, and the following observations will be noticed:
- Case 6: If the m-th failure time occurs before time , , then all of the remaining operating units =n−m− are eliminated from the experiment, thereby stopping the experiment at ; see Figure 7. In this case, , , and the following observations will be noticed:
3. Estimation Methods
3.1. Maximum Likelihood Estimation
- For Cases 1–5:
- For Case 6:
- For Cases 1–5, the MLEs of and can be calculated by equating to zero the first partial derivatives of (20) with respect to and . Then, the likelihood equations take the formsFrom (22), the MLE of can be obtained as a function of as follows:By substituting in (23) and solving the likelihood equation with respect to by using any numerical iteration method, the MLE of can be obtained.
- For Case 6, no failures were detected under accelerated stress conditions, so the MLE of does not exist, but the MLE of can be calculated by equating to zero the first partial derivative of (21) with respect to . Then, the MLE of is given by
- In Cases 1,2, and 4, if , no failures were detected under normal stress conditions; then, the MLE of β does not exist.
- In Case 3, if , no failures were detected under accelerated stress conditions; then, the MLE of α does not exist.
3.2. Percentile Estimation
- For Cases 1–5: It is possible to obtain the PEs and of and by minimizing the next quantity with respect to and .
- Case A:
- Case B:
- Case C:
whereMinimizing the quantity can be done by solving and with respect to and . - For Case 6, the PE of does not exist, but the PE of can be obtained by minimizing the next quantity with respect to :
4. Common Optimality Criteria
- A-optimality criterionThis criterion is concerned with the maximization of the trace of the FIM, which provides an overall information measure based on the marginal information about the parameters. It is preferred when the estimates are, at most, moderately correlated. Therefore, the optimal value of can be obtained by maximizing the trace of the FIM (29) of MLEs and , i.e.,
- D-optimality criterionThis criterion is concerned with maximizing the determinant of the FIM, which provides an overall measure of variability by taking into account the correlation between the estimates. It is preferred that it is applied when the estimates are highly correlated. It is also pertinent for the construction of the joint confidence region for the parameters. Therefore, the optimal value of can be obtained by maximizing (minimizing) the determinant of the FIM (inverse of the FIM) ((29) and (28)) of the MLEs and , i.e.,
5. Real Dataset
6. Simulation Study
- Assign values of the sample size n, stress change time , and parameters , and assign values of , and .
- Generate n observations from the Uniform(0,1) distribution.
- Apply the generalized type-I progressive hybrid censoring to the random sample obtained in Step 3, as shown in Section 2.1.
- Find the value of r, which is the number of observations with values , as follows:
- Compute the MLEs, PEs, NACIs, and LTCIs of , as shown in Section 3.
- Compute the optimal values, and , of the stress change time, as shown in Section 4.
- Replicate Steps 2–7 (= 5000) times.
- Calculate the mean of the estimates, mean squared error (MSE), and relative absolute bias (RAB) of over samples as follows:
- Using Step 9, calculate the mean of the estimates of the parameters with their MSEs and RABs.
- Calculate the average of the RABs (ARAB) and MSEs (AMSE) as follows:
- Evaluate the 95% NACIs and LTCIs of the parameters , and then evaluate their average interval lengths (AILs) and coverage probabilities (COVPs). In addition, also evaluate the average of the AILs (AAIL) as follows:
- CS1:
- CS2:
- CS3:
6.1. Simulation Results
- The MLEs are better than the PEs through the ARABs and AMSEs.
- The NACIs are better than the LTCIs through the AAILs.
- The optimal values and are close one to each other each time for Cases A and B. However, for Case C, the optimal value is less than and approaches the proposed value .
- For fixed values of m and , by increasing n, the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
- For fixed values of n and , by increasing m, the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
- For fixed values of n and m, by increasing , the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
- By increasing , or , the COVPs are close to 95% for Cases A and B. However, for Case C, the COVPs are higher than 95% because greater values of the MSEs of the estimates than those for Cases A and B were obtained, which led to wider CIs
- For fixed values of n and , by increasing m, the optimal values and increase.
- For fixed m and , by increasing n, the optimal values and decrease.
- For fixed n and m, by increasing , the optimal values and increase.
7. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Case 1 | Cases 2 and 3 | Cases 4 and 5 | |
---|---|---|---|
k | D | m | |
0 | 1 | 0 | |
Normal stress | 12.07, 14.0, 17.95, 19.5, 22.1, 23.11, 24.0, 24.0, 25.1, 26.46, 26.58, 26.9, 28.06, 34.0, 36.13, 36.64, |
(before ) | 40.85, 41.11, 42.63, 44.1, 46.3, 52.51, 54.0, 58.09, 62.68, 64.17, 72.25, 73.13, 83.63, 86.9, 90.09, |
91.22, 91.56, 94.38 | |
Accelerated stress | 97.71, 101.53, 102.1, 105.1, 105.11, 109.2, 112.11, 114.4, 117.9, 119.58, 120.2, |
(after ) | 121.9, 122.5, 123.6, 126.5, 126.95, 129.25, 130.1, 136.31 |
Parameters | K-S | p-Value | |
---|---|---|---|
Case A | 0.0797349 | 0.88902758 | |
Case B | 0.0866267 | 0.82128778 | |
Case C | 0.0812119 | 0.87567578 |
Scheme I | |
---|---|
Normal stress | 12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.0, 25.10, 26.46, 26.58, 26.90, 28.06, 34.00, |
36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25, | |
73.13, 83.63, 86.90, 90.09, 91.22, 91.56, 94.38 | |
Accelerated stress | 97.71, 101.53, 102.1, 105.1, 105.11, 109.2, 112.11, 114.40 |
() | |
Scheme II | |
Normal stress | 12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.00, 25.10, 26.46, 26.58, 26.9, 28.06, 34.00, |
36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25, | |
73.13, 83.63, 86.9, 90.09, 91.22, 91.56, 94.38 | |
Accelerated stress | 97.71, 101.53, 102.10, 105.10, 105.11, 109.20, 112.11, 114.40, 117.90, 119.58, 120.20, |
() | 121.90, 122.50, 123.60 |
Scheme III | |
Normal stress | 12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.00, 25.10, 26.46, 26.58, 26.9, 28.06, 34.00, |
36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25, | |
73.13, 83.63, 86.9, 90.09, 91.22, 91.56, 94.38 | |
Accelerated stress | 97.71, 101.53, 102.10, 105.10, 105.11, 109.20, 112.11, 114.40, 117.90, 119.58, 120.20, |
() | 121.90, 122.50, 123.60, 126.50, 126.95, 129.25, 130.10, 136.31 |
PE() | MLE() | NACI() | LTCI() | |||
---|---|---|---|---|---|---|
PE() | MLE() | NACI() | LTCI() | |||
Case A | 116 | 0.45343 | 0.30888 | (0.0, 0.63271) | (0.10826, 0.88126) | 71.3397 |
133.038 | 130.799 | (87.3089, 174.288) | (93.8, 182.391) | 71.3397 | ||
125 | 0.40396 | 0.39898 | (0.17329, 0.62467) | (0.22661, 0.70245) | 71.3397 | |
132.899 | 132.476 | (88.3358, 176.616) | (94.9364, 184.859) | 71.3397 | ||
140 | 0.42124 | 0.40638 | (0.21944, 0.59332) | (0.25654, 0.64374) | 71.3397 | |
133.063 | 132.597 | (88.551, 176.642) | (95.1195, 184.84) | 71.3397 | ||
Case B | 116 | 0.11873 | 0.05355 | (0.0, 0.11652) | (0.016528, 0.17353) | 114.783 |
132.822 | 127.842 | (86.3012, 169.382) | (92.375, 176.926) | 114.881 | ||
125 | 0.07414 | 0.05983 | (0.02756, 0.09209) | (0.03489, 0.10259) | 122.951 | |
132.046 | 128.837 | (87.345, 170.329) | (93.3633, 177.79) | 123.277 | ||
140 | 0.06431 | 0.04765 | (0.02615, 0.06915) | (0.03035, 0.07482) | 131.446 | |
131.521 | 125.266 | (85.9392, 164.593) | (91.5141, 171.466) | 132.369 | ||
Case C | 116 | 0.42073 | 0.22842 | (0.0, 0.83665) | (0.01593, 3.27483) | 97.3663 |
131.48 | 131.674 | (87.1202, 176.227) | (93.875, 184.692) | 97.4677 | ||
125 | 0.37920 | 0.39619 | (0.0, 1.24462) | (0.046546, 3.37232) | 97.1638 | |
131.313 | 131.282 | (87.3779, 175.186) | (93.9649, 183.419) | 97.1568 | ||
140 | 0.53392 | 0.45523 | (0.0, 1.36748) | (0.06137, 3.37696) | 97.212 | |
132.409 | 130.959 | (87.365, 174.554) | (93.8786, 182.686) | 97.206 |
PE | MLE | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
MSE() | RAB() | AMSE | MSE() | RAB() | AMSE | |||||||
n | m | k | CS | MSE() | RAB() | ARAB | MSE() | RAB() | ARAB | |||
30 | 18 | 12 | 0.75 | 1 | 4.10970 | 11.2033 | 1.30263 | 5.63093 | 3.37393 | 3.42300 | 0.70773 | 1.75625 |
0.64754 | 0.05853 | 0.29228 | 0.79746 | 0.70101 | 0.08950 | 0.37164 | 0.53968 | |||||
2 | 4.33833 | 12.4739 | 1.38312 | 6.26307 | 3.47865 | 3.93665 | 0.75631 | 1.99988 | ||||
0.64156 | 0.05224 | 0.27201 | 0.82757 | 0.67197 | 0.06311 | 0.31505 | 0.53568 | |||||
3 | 4.19268 | 11.1990 | 1.30426 | 5.62179 | 3.56698 | 4.29929 | 0.79475 | 2.17842 | ||||
0.63390 | 0.04462 | 0.26054 | 0.78240 | 0.66843 | 0.05754 | 0.30622 | 0.55049 | |||||
1.25 | 1 | 2.68433 | 2.59585 | 0.59042 | 1.33347 | 2.84073 | 2.30459 | 0.5663 | 1.18466 | |||
0.63422 | 0.07108 | 0.32037 | 0.45539 | 0.65492 | 0.06473 | 0.31353 | 0.43992 | |||||
2 | 2.76357 | 2.94032 | 0.61863 | 1.49434 | 2.97708 | 2.92565 | 0.62315 | 1.48793 | ||||
0.62260 | 0.04835 | 0.28770 | 0.45316 | 0.63643 | 0.05022 | 0.28241 | 0.45278 | |||||
3 | 2.8303 | 3.23319 | 0.66009 | 1.6401 | 3.01581 | 3.08544 | 0.63716 | 1.56479 | ||||
0.62345 | 0.04701 | 0.28397 | 0.47203 | 0.63185 | 0.04414 | 0.27165 | 0.45441 | |||||
24 | 12 | 0.75 | 1 | 2.97613 | 3.89284 | 0.75303 | 1.97052 | 2.95832 | 2.20531 | 0.55766 | 1.13000 | |
0.61994 | 0.04820 | 0.27166 | 0.51234 | 0.65119 | 0.05468 | 0.28791 | 0.42278 | |||||
2 | 2.95310 | 3.74576 | 0.73898 | 1.89329 | 3.02478 | 2.44235 | 0.58229 | 1.24426 | ||||
0.60994 | 0.04081 | 0.24948 | 0.49423 | 0.64388 | 0.04617 | 0.26949 | 0.42589 | |||||
3 | 2.95322 | 3.63054 | 0.73559 | 1.83620 | 2.94569 | 2.12297 | 0.54747 | 1.08389 | ||||
0.61535 | 0.04187 | 0.26235 | 0.49897 | 0.64359 | 0.04482 | 0.27197 | 0.40972 | |||||
1.25 | 1 | 2.55345 | 1.79452 | 0.49628 | 0.91790 | 2.61782 | 1.65911 | 0.49061 | 0.85252 | |||
0.61176 | 0.04127 | 0.26281 | 0.37955 | 0.62809 | 0.04592 | 0.27096 | 0.38078 | |||||
2 | 2.53464 | 1.82050 | 0.49915 | 0.92836 | 2.66263 | 1.85759 | 0.50809 | 0.94796 | ||||
0.60397 | 0.03621 | 0.24803 | 0.37359 | 0.61612 | 0.03833 | 0.25178 | 0.37993 | |||||
3 | 2.50760 | 1.65639 | 0.48453 | 0.84743 | 2.56043 | 1.54207 | 0.47052 | 0.78880 | ||||
0.60903 | 0.03848 | 0.25598 | 0.37026 | 0.61692 | 0.03554 | 0.24879 | 0.35965 | |||||
30 | – | – | – | 2.30212 | 1.24452 | 0.41972 | 0.63620 | 2.42642 | 1.15576 | 0.41019 | 0.59154 | |
0.58749 | 0.02788 | 0.22429 | 0.32200 | 0.59864 | 0.02731 | 0.21961 | 0.31490 | |||||
60 | 36 | 24 | 0.75 | 1 | 3.43922 | 7.39602 | 1.01964 | 3.71142 | 2.53677 | 1.29372 | 0.43634 | 0.66028 |
0.60888 | 0.02682 | 0.20950 | 0.61457 | 0.60662 | 0.02685 | 0.21441 | 0.32537 | |||||
2 | 2.64437 | 2.32974 | 0.58995 | 1.17467 | 2.60190 | 1.41379 | 0.45254 | 0.71662 | ||||
0.58736 | 0.01959 | 0.18838 | 0.38916 | 0.59668 | 0.01944 | 0.18739 | 0.31997 | |||||
3 | 2.70948 | 2.58468 | 0.62010 | 1.30140 | 2.66926 | 1.54932 | 0.47151 | 0.78424 | ||||
0.58763 | 0.01812 | 0.18414 | 0.40212 | 0.59808 | 0.01916 | 0.18572 | 0.32861 | |||||
1.25 | 1 | 2.33366 | 1.01848 | 0.38467 | 0.52067 | 2.33813 | 0.96605 | 0.37642 | 0.49470 | |||
0.58622 | 0.02286 | 0.20352 | 0.29410 | 0.59120 | 0.02335 | 0.20379 | 0.29010 | |||||
2 | 2.40504 | 1.12983 | 0.40460 | 0.57439 | 2.37395 | 1.06426 | 0.39390 | 0.54091 | ||||
0.58864 | 0.01895 | 0.18748 | 0.29604 | 0.58188 | 0.01756 | 0.18222 | 0.28806 | |||||
3 | 2.43026 | 1.19981 | 0.41538 | 0.60900 | 2.43005 | 1.19989 | 0.41316 | 0.60876 | ||||
0.58722 | 0.01819 | 0.18599 | 0.30068 | 0.58375 | 0.01763 | 0.18145 | 0.29731 | |||||
48 | 24 | 0.75 | 1 | 2.39614 | 1.54233 | 0.49143 | 0.78028 | 2.34300 | 0.97986 | 0.38883 | 0.49868 | |
0.58246 | 0.01823 | 0.18235 | 0.33689 | 0.58855 | 0.01750 | 0.17509 | 0.28196 | |||||
2 | 2.42587 | 1.57380 | 0.49546 | 0.79447 | 2.35547 | 0.97236 | 0.38550 | 0.49363 | ||||
0.57768 | 0.01514 | 0.16779 | 0.33163 | 0.58533 | 0.01490 | 0.16574 | 0.27562 | |||||
3 | 2.39619 | 1.54925 | 0.48748 | 0.78304 | 2.32671 | 0.91357 | 0.38036 | 0.46465 | ||||
0.58120 | 0.01682 | 0.17773 | 0.33260 | 0.58321 | 0.01573 | 0.17248 | 0.27642 | |||||
1.25 | 1 | 2.27817 | 0.73893 | 0.32926 | 0.37744 | 2.23473 | 0.69062 | 0.32600 | 0.35332 | |||
0.58029 | 0.01594 | 0.17331 | 0.25128 | 0.58016 | 0.01601 | 0.17269 | 0.24934 | |||||
2 | 2.27104 | 0.76043 | 0.33433 | 0.38763 | 2.25127 | 0.72031 | 0.32935 | 0.36714 | ||||
0.57929 | 0.01484 | 0.16856 | 0.25145 | 0.57680 | 0.01396 | 0.16228 | 0.24581 | |||||
3 | 2.25994 | 0.69591 | 0.32225 | 0.35576 | 2.22000 | 0.64545 | 0.31412 | 0.32984 | ||||
0.58023 | 0.01560 | 0.17289 | 0.24757 | 0.57510 | 0.01424 | 0.16552 | 0.23982 | |||||
60 | – | – | – | 2.15387 | 0.55668 | 0.28837 | 0.28452 | 2.20042 | 0.51189 | 0.27649 | 0.26211 | |
0.57066 | 0.01236 | 0.15467 | 0.22152 | 0.57498 | 0.01233 | 0.15267 | 0.21458 |
NACI | LTCI | Optimal Value | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
COVP | AIL | COVP | AIL | |||||||||
n | m | k | CS | COVP | AIL | AAIL | COVP | AIL | AAIL | |||
30 | 18 | 12 | 0.75 | 1 | 94.14 | 6.47228 | 3.76255 | 80.62 | 9.21429 | 5.18726 | 0.56556 | 0.57287 |
98.72 | 1.05283 | 97.12 | 1.16024 | |||||||||
2 | 92.14 | 6.56681 | 3.73274 | 78.30 | 9.28831 | 5.12874 | 0.48625 | 0.47228 | ||||
98.68 | 0.89867 | 96.74 | 0.96916 | |||||||||
3 | 91.42 | 6.69255 | 3.78226 | 77.18 | 9.41054 | 5.17359 | 0.47352 | 0.45495 | ||||
98.64 | 0.87197 | 97.06 | 0.93665 | |||||||||
1.25 | 1 | 95.06 | 5.04383 | 2.98203 | 84.28 | 6.93378 | 3.9673 | 0.58531 | 0.60134 | |||
98.48 | 0.92023 | 97.34 | 1.00083 | |||||||||
2 | 92.62 | 5.24578 | 3.03094 | 81.82 | 7.16377 | 4.01912 | 0.51999 | 0.51414 | ||||
98.18 | 0.81611 | 96.90 | 0.87447 | |||||||||
3 | 92.86 | 5.38945 | 3.09228 | 81.96 | 7.41417 | 4.13191 | 0.5002 | 0.49022 | ||||
97.92 | 0.79511 | 97.1 | 0.84964 | |||||||||
24 | 12 | 0.75 | 1 | 94.98 | 5.59843 | 3.21471 | 84.56 | 8.13174 | 4.51119 | 0.53359 | 0.53728 | |
98.38 | 0.83100 | 96.84 | 0.89064 | |||||||||
2 | 93.7 | 5.64222 | 3.21152 | 83.26 | 8.11909 | 4.47503 | 0.49546 | 0.49248 | ||||
98.48 | 0.78083 | 96.72 | 0.83096 | |||||||||
3 | 95.36 | 5.59081 | 3.19587 | 85.04 | 8.07499 | 4.465 | 0.53904 | 0.53693 | ||||
98.48 | 0.80092 | 97.1 | 0.85502 | |||||||||
1.25 | 1 | 93.90 | 4.32257 | 2.53875 | 84.90 | 5.72211 | 3.26230 | 0.57303 | 0.58748 | |||
97.106 | 0.75494 | 96.10 | 0.80248 | |||||||||
2 | 93.64 | 4.41189 | 2.56065 | 84.02 | 5.83504 | 3.29259 | 0.5255 | 0.53127 | ||||
97.00 | 0.70941 | 96.26 | 0.75014 | |||||||||
3 | 95.00 | 4.30250 | 2.51391 | 87.02 | 5.75165 | 3.26019 | 0.57194 | 0.57988 | ||||
97.18 | 0.72532 | 97.06 | 0.76872 | |||||||||
30 | – | – | – | 95.18 | 3.80526 | 2.21832 | 88.00 | 4.73587 | 2.69875 | 0.56485 | 0.57587 | |
97.06 | 0.63138 | 96.68 | 0.66162 | |||||||||
60 | 36 | 24 | 0.75 | 1 | 95.22 | 4.60444 | 2.60893 | 88.54 | 6.66823 | 3.65434 | 0.52015 | 0.52334 |
97.76 | 0.61343 | 96.78 | 0.64044 | |||||||||
2 | 94.76 | 4.68530 | 2.61318 | 87.06 | 6.74321 | 3.65160 | 0.44714 | 0.44359 | ||||
97.66 | 0.54107 | 96.32 | 0.56000 | |||||||||
3 | 94.34 | 4.77857 | 2.65535 | 86.54 | 6.82727 | 3.68865 | 0.45700 | 0.44899 | ||||
97.32 | 0.53212 | 96.28 | 0.55003 | |||||||||
1.25 | 1 | 94.86 | 3.51239 | 2.03809 | 88.58 | 4.30683 | 2.44629 | 0.56400 | 0.57334 | |||
96.80 | 0.56379 | 96.24 | 0.58575 | |||||||||
2 | 94.80 | 3.60637 | 2.05704 | 88.28 | 4.48194 | 2.50303 | 0.48178 | 0.48230 | ||||
95.76 | 0.50771 | 95.98 | 0.52412 | |||||||||
3 | 94.24 | 3.70235 | 2.10175 | 86.90 | 4.60024 | 2.55853 | 0.48829 | 0.48411 | ||||
96.02 | 0.50114 | 95.12 | 0.51681 | |||||||||
48 | 24 | 0.75 | 1 | 96.04 | 4.03729 | 2.2744 | 89.80 | 5.71223 | 3.12012 | 0.51759 | 0.52259 | |
97.32 | 0.51152 | 96.34 | 0.52801 | |||||||||
2 | 95.86 | 4.07491 | 2.27961 | 90.28 | 5.74677 | 3.12258 | 0.48015 | 0.48366 | ||||
97.94 | 0.48431 | 96.80 | 0.49840 | |||||||||
3 | 96.54 | 4.03397 | 2.2648 | 90.72 | 5.70665 | 3.10876 | 0.51893 | 0.52096 | ||||
96.86 | 0.49563 | 96.56 | 0.51087 | |||||||||
1.25 | 1 | 95.62 | 3.04419 | 1.76014 | 89.98 | 3.52464 | 2.00718 | 0.56559 | 0.57375 | |||
95.96 | 0.47609 | 95.68 | 0.48972 | |||||||||
2 | 95.02 | 3.08273 | 1.7684 | 89.84 | 3.57207 | 2.01903 | 0.52023 | 0.52555 | ||||
95.60 | 0.45407 | 95.48 | 0.46600 | |||||||||
3 | 96.10 | 3.04957 | 1.75644 | 91.22 | 3.52236 | 1.99921 | 0.56692 | 0.57116 | ||||
96.18 | 0.4633 | 95.98 | 0.47606 | |||||||||
60 | – | – | – | 95.24 | 2.68636 | 1.55267 | 91.24 | 2.93992 | 1.68416 | 0.56996 | 0.57573 | |
95.68 | 0.41897 | 94.88 | 0.42839 |
PE | MLE | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
MSE() | RAB() | AMSE | MSE() | RAB() | AMSE | |||||||
n | m | k | CS | MSE() | RAB() | ARAB | MSE() | RAB() | ARAB | |||
30 | 18 | 12 | 0.75 | 1 | 2.82931 | 3.54341 | 0.80975 | 1.78568 | 2.90194 | 2.37936 | 0.60538 | 1.22738 |
0.59976 | 0.02794 | 0.22394 | 0.51684 | 0.68174 | 0.07540 | 0.34360 | 0.47449 | |||||
2 | 3.16299 | 5.37988 | 0.90301 | 2.71953 | 2.99101 | 2.78859 | 0.63979 | 1.42086 | ||||
0.64115 | 0.05919 | 0.29807 | 0.60054 | 0.65172 | 0.05313 | 0.29298 | 0.46639 | |||||
3 | 3.13278 | 4.77410 | 0.86444 | 2.41437 | 3.03398 | 2.95214 | 0.65726 | 1.49995 | ||||
0.62655 | 0.05464 | 0.28358 | 0.57401 | 0.64625 | 0.04777 | 0.28181 | 0.46954 | |||||
1.25 | 1 | 2.39827 | 1.93518 | 0.52975 | 0.99867 | 2.56056 | 1.83094 | 0.51694 | 0.94684 | |||
0.6198 | 0.06216 | 0.30402 | 0.41688 | 0.63873 | 0.06274 | 0.31018 | 0.41356 | |||||
2 | 2.49453 | 2.23036 | 0.55891 | 1.13844 | 2.62990 | 2.17056 | 0.55417 | 1.10792 | ||||
0.61438 | 0.04651 | 0.28040 | 0.41965 | 0.62341 | 0.04528 | 0.27137 | 0.41277 | |||||
3 | 2.51084 | 2.35251 | 0.58018 | 1.19797 | 2.67482 | 2.30001 | 0.56872 | 1.17173 | ||||
0.61479 | 0.04343 | 0.27009 | 0.42513 | 0.62354 | 0.04346 | 0.2708 | 0.41976 | |||||
24 | 12 | 0.75 | 1 | 2.66153 | 2.84045 | 0.67903 | 1.44386 | 2.55870 | 1.51688 | 0.48784 | 0.78104 | |
0.61423 | 0.04727 | 0.26993 | 0.47448 | 0.63323 | 0.04519 | 0.26975 | 0.37880 | |||||
2 | 2.65765 | 2.80670 | 0.66891 | 1.42267 | 2.57206 | 1.52838 | 0.48807 | 0.78438 | ||||
0.60499 | 0.03863 | 0.24942 | 0.45916 | 0.62852 | 0.04037 | 0.25546 | 0.37176 | |||||
3 | 2.66155 | 2.70155 | 0.66215 | 1.37114 | 2.54656 | 1.48692 | 0.47915 | 0.76343 | ||||
0.60974 | 0.04073 | 0.25680 | 0.45948 | 0.62569 | 0.03995 | 0.25838 | 0.36877 | |||||
1.25 | 1 | 2.27670 | 1.27666 | 0.43552 | 0.65768 | 2.34350 | 1.13894 | 0.42183 | 0.58874 | |||
0.60144 | 0.03869 | 0.25497 | 0.34525 | 0.61286 | 0.03854 | 0.25427 | 0.33805 | |||||
2 | 2.25654 | 1.23947 | 0.43502 | 0.63557 | 2.37776 | 1.24784 | 0.43577 | 0.64098 | ||||
0.59460 | 0.03167 | 0.23929 | 0.33716 | 0.60409 | 0.03411 | 0.24156 | 0.33866 | |||||
3 | 2.26794 | 1.2673 | 0.42851 | 0.65210 | 2.34428 | 1.15663 | 0.42587 | 0.59623 | ||||
0.60239 | 0.0369 | 0.25062 | 0.33956 | 0.60774 | 0.03583 | 0.24687 | 0.33637 | |||||
30 | – | – | – | 2.16628 | 0.93180 | 0.37558 | 0.47932 | 2.25434 | 0.85314 | 0.36844 | 0.44084 | |
0.58901 | 0.02684 | 0.21922 | 0.29740 | 0.59955 | 0.02853 | 0.22622 | 0.29733 | |||||
60 | 36 | 24 | 0.75 | 1 | 2.33399 | 1.57484 | 0.51583 | 0.79989 | 2.25629 | 0.91773 | 0.39724 | 0.47061 |
0.58825 | 0.02495 | 0.20946 | 0.36264 | 0.59751 | 0.02348 | 0.20279 | 0.30001 | |||||
2 | 2.43163 | 1.79258 | 0.54229 | 0.90620 | 2.2866 | 0.97781 | 0.40555 | 0.49762 | ||||
0.58731 | 0.01982 | 0.18985 | 0.36607 | 0.58566 | 0.01742 | 0.18139 | 0.29347 | |||||
3 | 2.45710 | 1.81181 | 0.54569 | 0.91491 | 2.33607 | 1.08403 | 0.42181 | 0.55021 | ||||
0.58494 | 0.01802 | 0.18516 | 0.36543 | 0.58618 | 0.01639 | 0.17513 | 0.29847 | |||||
1.25 | 1 | 2.14646 | 0.71289 | 0.33572 | 0.36726 | 2.15126 | 0.67416 | 0.33218 | 0.34784 | |||
0.58273 | 0.02164 | 0.19793 | 0.26682 | 0.58502 | 0.02153 | 0.19829 | 0.26523 | |||||
2 | 2.18068 | 0.83142 | 0.35672 | 0.42469 | 2.18173 | 0.75843 | 0.34926 | 0.38770 | ||||
0.58181 | 0.01796 | 0.18432 | 0.27052 | 0.57677 | 0.01697 | 0.18145 | 0.26536 | |||||
3 | 2.18179 | 0.80429 | 0.35137 | 0.41070 | 2.21906 | 0.83871 | 0.36420 | 0.42749 | ||||
0.58197 | 0.01710 | 0.17881 | 0.26509 | 0.57718 | 0.01627 | 0.17642 | 0.27031 | |||||
48 | 24 | 0.75 | 1 | 2.20066 | 1.11075 | 0.43534 | 0.56389 | 2.09036 | 0.67092 | 0.34587 | 0.34366 | |
0.57946 | 0.01702 | 0.17808 | 0.30671 | 0.57909 | 0.01639 | 0.17340 | 0.25963 | |||||
2 | 2.21804 | 1.16005 | 0.44356 | 0.58810 | 2.12635 | 0.70902 | 0.35412 | 0.36170 | ||||
0.57796 | 0.01614 | 0.17109 | 0.30732 | 0.57727 | 0.01438 | 0.16389 | 0.25900 | |||||
3 | 2.22030 | 1.08627 | 0.43096 | 0.55127 | 2.10469 | 0.67264 | 0.34647 | 0.34364 | ||||
0.57881 | 0.01627 | 0.17439 | 0.30268 | 0.57773 | 0.01464 | 0.16522 | 0.25585 | |||||
1.25 | 1 | 2.09235 | 0.46897 | 0.27601 | 0.24186 | 2.08515 | 0.46462 | 0.27725 | 0.23961 | |||
0.57649 | 0.01475 | 0.16596 | 0.22099 | 0.57627 | 0.01460 | 0.16689 | 0.22207 | |||||
2 | 2.10153 | 0.52980 | 0.28948 | 0.27176 | 2.10875 | 0.53386 | 0.29658 | 0.27428 | ||||
0.57430 | 0.01372 | 0.16352 | 0.22650 | 0.57566 | 0.01469 | 0.16694 | 0.23176 | |||||
3 | 2.07216 | 0.48080 | 0.27873 | 0.24758 | 2.08643 | 0.46544 | 0.27737 | 0.23989 | ||||
0.57299 | 0.01435 | 0.16614 | 0.22243 | 0.57450 | 0.01434 | 0.16532 | 0.22135 | |||||
60 | – | – | – | 2.03842 | 0.38016 | 0.25164 | 0.19590 | 2.04836 | 0.35345 | 0.24393 | 0.18223 | |
0.56937 | 0.01163 | 0.15094 | 0.20129 | 0.57014 | 0.01100 | 0.14767 | 0.19580 |
NACI | LTCI | Optimal Value | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
COVP | AIL | COVP | AIL | |||||||||
n | m | k | CS | COVP | AIL | AAIL | COVP | AIL | AAIL | |||
30 | 18 | 12 | 0.75 | 1 | 93.16 | 5.35176 | 3.17217 | 81.8 | 7.70284 | 4.39454 | 0.53412 | 0.53472 |
98.32 | 0.99258 | 97.12 | 1.08625 | |||||||||
2 | 91.92 | 5.47946 | 3.16522 | 80.8 | 7.83493 | 4.37456 | 0.49218 | 0.46833 | ||||
98.16 | 0.85099 | 96.84 | 0.9142 | |||||||||
3 | 91.62 | 5.58575 | 3.20481 | 80.02 | 8.0354 | 4.45869 | 0.49094 | 0.46331 | ||||
98.16 | 0.82387 | 96.78 | 0.88198 | |||||||||
1.25 | 1 | 93.16 | 4.21141 | 2.5428 | 84.4 | 5.4724 | 3.20957 | 0.58178 | 0.58047 | |||
96.9 | 0.87419 | 96.68 | 0.94673 | |||||||||
2 | 92.56 | 4.36677 | 2.57477 | 83.24 | 5.70721 | 3.27176 | 0.53877 | 0.51030 | ||||
97.42 | 0.78277 | 96.48 | 0.8363 | |||||||||
3 | 93.02 | 4.49523 | 2.63212 | 83.34 | 5.87499 | 3.34729 | 0.53431 | 0.50136 | ||||
97.54 | 0.76901 | 96.92 | 0.8196 | |||||||||
24 | 12 | 0.75 | 1 | 94.44 | 4.63114 | 2.70773 | 86.02 | 6.72956 | 3.78325 | 0.53800 | 0.53265 | |
97.46 | 0.78432 | 96.86 | 0.83694 | |||||||||
2 | 94.58 | 4.66790 | 2.70536 | 85.46 | 6.76652 | 3.77721 | 0.51335 | 0.49896 | ||||
97.56 | 0.74282 | 96.84 | 0.78790 | |||||||||
3 | 94.28 | 4.62708 | 2.6923 | 86.28 | 6.71204 | 3.75891 | 0.5393 | 0.52938 | ||||
97.6 | 0.75751 | 96.84 | 0.80579 | |||||||||
1.25 | 1 | 94.10 | 3.56791 | 2.14128 | 86.74 | 4.38774 | 2.57227 | 0.59115 | 0.58196 | |||
96.78 | 0.71464 | 96.06 | 0.75681 | |||||||||
2 | 93.90 | 3.6468 | 2.16209 | 86.30 | 4.51675 | 2.61545 | 0.56063 | 0.53988 | ||||
96.16 | 0.67738 | 96.16 | 0.71415 | |||||||||
3 | 94.54 | 3.57774 | 2.13757 | 86.44 | 4.4192 | 2.57818 | 0.59352 | 0.57819 | ||||
96.3 | 0.6974 | 96.54 | 0.73717 | |||||||||
30 | – | – | – | 94.62 | 3.16431 | 1.8919 | 87.94 | 3.70022 | 2.1741 | 0.59591 | 0.58135 | |
96.42 | 0.6195 | 96.10 | 0.64797 | |||||||||
60 | 36 | 24 | 0.75 | 1 | 95.52 | 3.83408 | 2.21163 | 89.14 | 5.38558 | 2.99967 | 0.54185 | 0.53177 |
97.18 | 0.58918 | 96.60 | 0.61376 | |||||||||
2 | 95.38 | 3.91135 | 2.21626 | 88.58 | 5.55468 | 3.04669 | 0.47757 | 0.45583 | ||||
97.12 | 0.52116 | 96.38 | 0.53869 | |||||||||
3 | 94.62 | 3.98578 | 2.24866 | 87.78 | 5.65008 | 3.08907 | 0.48471 | 0.45998 | ||||
96.8 | 0.51153 | 96.48 | 0.52806 | |||||||||
1.25 | 1 | 94.70 | 2.872 | 1.70744 | 89.62 | 3.31451 | 1.93868 | 0.60036 | 0.58081 | |||
95.36 | 0.54288 | 95.34 | 0.56286 | |||||||||
2 | 94.68 | 2.98105 | 1.73734 | 88.86 | 3.45964 | 1.9843 | 0.53218 | 0.49806 | ||||
95.38 | 0.49363 | 95.2 | 0.50896 | |||||||||
3 | 94.34 | 3.06493 | 1.77565 | 88.24 | 3.586 | 2.0435 | 0.53529 | 0.49806 | ||||
95.36 | 0.48637 | 95.4 | 0.501 | |||||||||
48 | 24 | 0.75 | 1 | 96.60 | 3.36755 | 1.92931 | 91.70 | 4.54549 | 2.5258 | 0.54106 | 0.5308 | |
96.52 | 0.49107 | 96.46 | 0.50611 | |||||||||
2 | 95.88 | 3.40287 | 1.93511 | 90.76 | 4.59372 | 2.53703 | 0.5085 | 0.49354 | ||||
96.24 | 0.46735 | 96.02 | 0.48035 | |||||||||
3 | 96.42 | 3.3696 | 1.92491 | 91.62 | 4.5236 | 2.50896 | 0.54301 | 0.52961 | ||||
96.48 | 0.48022 | 96.4 | 0.49431 | |||||||||
1.25 | 1 | 94.66 | 2.47636 | 1.4684 | 91.14 | 2.70883 | 1.59087 | 0.60478 | 0.58173 | |||
95.38 | 0.46044 | 95.34 | 0.47291 | |||||||||
2 | 93.82 | 2.5207 | 1.48232 | 89.8 | 2.77381 | 1.61447 | 0.56629 | 0.53803 | ||||
94.78 | 0.44394 | 94.8 | 0.45513 | |||||||||
3 | 94.48 | 2.48398 | 1.46812 | 90.6 | 2.72151 | 1.59283 | 0.60687 | 0.58005 | ||||
95.12 | 0.45226 | 95.26 | 0.46414 | |||||||||
60 | – | – | – | 95.00 | 2.18996 | 1.29749 | 90.96 | 2.32862 | 1.37114 | 0.60744 | 0.58175 | |
95.48 | 0.40501 | 95.44 | 0.41366 |
PE | MLE | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
MSE() | RAB() | AMSE | MSE() | RAB() | AMSE | |||||||
n | m | k | CS | MSE() | RAB() | ARAB | MSE() | RAB() | ARAB | |||
30 | 18 | 12 | 0.75 | 1 | 2.17114 | 3.99222 | 1.49238 | 2.02063 | 1.56840 | 1.55187 | 0.97416 | 0.78835 |
0.66173 | 0.04904 | 0.30438 | 0.89838 | 0.58370 | 0.02484 | 0.21509 | 0.59462 | |||||
2 | 2.34821 | 4.58109 | 1.62639 | 2.31543 | 1.54813 | 1.50802 | 0.95967 | 0.76442 | ||||
0.67595 | 0.04977 | 0.30681 | 0.96660 | 0.56876 | 0.02081 | 0.20241 | 0.58104 | |||||
3 | 2.35557 | 4.68525 | 1.65054 | 2.36836 | 1.57068 | 1.52619 | 0.96485 | 0.77299 | ||||
0.68716 | 0.05148 | 0.31169 | 0.98111 | 0.56929 | 0.01980 | 0.19607 | 0.58046 | |||||
1.25 | 1 | 1.41464 | 1.80542 | 0.95059 | 0.91389 | 1.44685 | 1.31165 | 0.88403 | 0.66620 | |||
0.53475 | 0.02235 | 0.21540 | 0.58300 | 0.56785 | 0.02075 | 0.20141 | 0.54272 | |||||
2 | 1.4515 | 1.93496 | 0.97806 | 0.97717 | 1.47440 | 1.35351 | 0.89526 | 0.68699 | ||||
0.52402 | 0.01938 | 0.2025 | 0.59028 | 0.56890 | 0.02047 | 0.19854 | 0.54690 | |||||
3 | 1.49695 | 2.12906 | 1.02507 | 1.07402 | 1.48678 | 1.40529 | 0.91533 | 0.71242 | ||||
0.52636 | 0.01899 | 0.20086 | 0.61297 | 0.56449 | 0.01955 | 0.19408 | 0.55471 | |||||
24 | 12 | 0.75 | 1 | 2.11856 | 3.84570 | 1.45668 | 1.94282 | 1.52749 | 1.47995 | 0.94928 | 0.74896 | |
0.63982 | 0.03994 | 0.26919 | 0.86293 | 0.57299 | 0.01796 | 0.18741 | 0.56834 | |||||
2 | 1.97300 | 3.38275 | 1.34404 | 1.70325 | 1.53808 | 1.47733 | 0.94485 | 0.74736 | ||||
0.57566 | 0.02374 | 0.21203 | 0.77803 | 0.57166 | 0.01739 | 0.18555 | 0.56520 | |||||
3 | 1.95572 | 3.31454 | 1.32478 | 1.66900 | 1.52887 | 1.44759 | 0.93795 | 0.73276 | ||||
0.57791 | 0.02346 | 0.20947 | 0.76712 | 0.57408 | 0.01793 | 0.18908 | 0.56352 | |||||
1.25 | 1 | 1.38669 | 1.72566 | 0.91878 | 0.87199 | 1.42009 | 1.27118 | 0.87035 | 0.64421 | |||
0.54287 | 0.01832 | 0.19464 | 0.55671 | 0.56738 | 0.01724 | 0.18345 | 0.52690 | |||||
2 | 1.38908 | 1.73950 | 0.92703 | 0.87796 | 1.46111 | 1.36687 | 0.90733 | 0.69182 | ||||
0.53526 | 0.01641 | 0.18510 | 0.55606 | 0.56607 | 0.01676 | 0.18057 | 0.54395 | |||||
3 | 1.36414 | 1.71952 | 0.91591 | 0.86808 | 1.44570 | 1.33560 | 0.89572 | 0.67641 | ||||
0.53438 | 0.01665 | 0.18663 | 0.55127 | 0.56957 | 0.01722 | 0.18365 | 0.53969 | |||||
30 | – | – | – | 1.10612 | 1.15803 | 0.76992 | 0.58451 | 1.32290 | 1.14694 | 0.82315 | 0.58085 | |
0.51997 | 0.01100 | 0.15398 | 0.46195 | 0.56012 | 0.01477 | 0.17187 | 0.49751 | |||||
60 | 36 | 24 | 0.75 | 1 | 2.06335 | 3.70211 | 1.41565 | 1.86252 | 1.50620 | 1.40106 | 0.91983 | 0.70743 |
0.61422 | 0.02294 | 0.20854 | 0.81210 | 0.57217 | 0.01380 | 0.16241 | 0.54112 | |||||
2 | 2.25351 | 4.27479 | 1.56326 | 2.14736 | 1.49731 | 1.41194 | 0.92624 | 0.71181 | ||||
0.61433 | 0.01993 | 0.19675 | 0.88001 | 0.56619 | 0.01168 | 0.15238 | 0.53931 | |||||
3 | 1.93645 | 3.20929 | 1.31084 | 1.61172 | 1.49309 | 1.40126 | 0.91361 | 0.70647 | ||||
0.57022 | 0.01415 | 0.16772 | 0.73928 | 0.56567 | 0.01168 | 0.15344 | 0.53353 | |||||
1.25 | 1 | 1.34742 | 1.58772 | 0.88932 | 0.80006 | 1.37573 | 1.18773 | 0.83789 | 0.60038 | |||
0.54364 | 0.01239 | 0.15992 | 0.52462 | 0.56542 | 0.01303 | 0.16160 | 0.49974 | |||||
2 | 1.37444 | 1.67479 | 0.89721 | 0.84317 | 1.39712 | 1.23752 | 0.85905 | 0.62439 | ||||
0.53713 | 0.01155 | 0.15651 | 0.52686 | 0.56330 | 0.01125 | 0.15131 | 0.50518 | |||||
3 | 1.40157 | 1.71212 | 0.93292 | 0.86176 | 1.40659 | 1.26004 | 0.86067 | 0.63556 | ||||
0.53468 | 0.01140 | 0.15780 | 0.54536 | 0.56228 | 0.01107 | 0.14951 | 0.50509 | |||||
48 | 24 | 0.75 | 1 | 1.78336 | 2.73233 | 1.19009 | 1.37272 | 1.47256 | 1.32299 | 0.89085 | 0.66688 | |
0.57615 | 0.01312 | 0.15946 | 0.67477 | 0.56903 | 0.01077 | 0.14675 | 0.51880 | |||||
2 | 1.80859 | 2.77815 | 1.19801 | 1.39538 | 1.46585 | 1.33092 | 0.89158 | 0.67070 | ||||
0.57113 | 0.01261 | 0.15559 | 0.67680 | 0.56806 | 0.01049 | 0.14412 | 0.51785 | |||||
3 | 1.76336 | 2.64301 | 1.16230 | 1.32788 | 1.46537 | 1.35280 | 0.89503 | 0.68157 | ||||
0.57393 | 0.01276 | 0.15685 | 0.65957 | 0.56617 | 0.01034 | 0.14459 | 0.51981 | |||||
1.25 | 1 | 1.34385 | 1.49593 | 0.85893 | 0.75335 | 1.3559 | 1.14415 | 0.82289 | 0.57745 | |||
0.54792 | 0.01077 | 0.14955 | 0.50424 | 0.5642 | 0.01075 | 0.14599 | 0.48444 | |||||
2 | 1.35650 | 1.55408 | 0.86808 | 0.78207 | 1.35240 | 1.14350 | 0.81526 | 0.57663 | ||||
0.54656 | 0.01006 | 0.14439 | 0.50623 | 0.56313 | 0.00976 | 0.14033 | 0.47779 | |||||
3 | 1.32831 | 1.39706 | 0.83624 | 0.70369 | 1.36178 | 1.11986 | 0.81096 | 0.56509 | ||||
0.54477 | 0.01032 | 0.14578 | 0.49101 | 0.56434 | 0.01032 | 0.14414 | 0.47755 | |||||
60 | – | – | – | 1.08480 | 0.89976 | 0.69059 | 0.45346 | 1.29689 | 1.00335 | 0.75650 | 0.50599 | |
0.52858 | 0.00716 | 0.12373 | 0.40716 | 0.55882 | 0.00862 | 0.13227 | 0.44438 |
NACI | LTCI | Optimal Value | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
COVP | AIL | COVP | AIL | |||||||||
n | m | k | CS | COVP | AIL | AAIL | COVP | AIL | AAIL | |||
30 | 18 | 12 | 0.75 | 1 | 100.00 | 7.58446 | 4.22565 | 100.00 | 8.3369 | 4.64840 | 0.20424 | 0.44114 |
99.14 | 0.86685 | 99.96 | 0.95989 | |||||||||
2 | 99.98 | 7.64752 | 4.20315 | 99.98 | 8.34642 | 4.58494 | 0.19781 | 0.40666 | ||||
98.16 | 0.75877 | 99.78 | 0.82347 | |||||||||
3 | 99.96 | 7.75997 | 4.25047 | 99.98 | 8.46083 | 4.62971 | 0.20022 | 0.40418 | ||||
97.60 | 0.74097 | 99.50 | 0.79860 | |||||||||
1.25 | 1 | 100.00 | 6.38073 | 3.58809 | 100.00 | 6.77272 | 3.82209 | 0.217 | 0.47592 | |||
97.64 | 0.79546 | 99.68 | 0.87147 | |||||||||
2 | 100.00 | 6.54931 | 3.64013 | 100.00 | 6.90472 | 3.84629 | 0.20671 | 0.43776 | ||||
96.66 | 0.73094 | 99.36 | 0.78787 | |||||||||
3 | 100.00 | 6.53461 | 3.62206 | 100.00 | 6.95781 | 3.85946 | 0.20683 | 0.43329 | ||||
96.70 | 0.70950 | 99.30 | 0.76111 | |||||||||
24 | 12 | 0.75 | 1 | 100.00 | 7.25760 | 4.00007 | 100.00 | 7.69363 | 4.24681 | 0.20117 | 0.43564 | |
98.28 | 0.74254 | 99.60 | 0.79999 | |||||||||
2 | 100.00 | 7.28863 | 3.99804 | 100.00 | 7.74797 | 4.25314 | 0.19809 | 0.41539 | ||||
97.38 | 0.70745 | 99.32 | 0.75831 | |||||||||
3 | 100.00 | 7.22512 | 3.97734 | 100.00 | 7.66505 | 4.22436 | 0.20027 | 0.43711 | ||||
97.86 | 0.72955 | 99.42 | 0.78367 | |||||||||
1.25 | 1 | 99.98 | 5.98044 | 3.34178 | 99.98 | 6.38817 | 3.57219 | 0.21180 | 0.46854 | |||
97.84 | 0.70312 | 99.36 | 0.75622 | |||||||||
2 | 99.98 | 6.11328 | 3.39474 | 99.98 | 6.54616 | 3.63442 | 0.20548 | 0.44463 | ||||
97.52 | 0.67620 | 99.06 | 0.72267 | |||||||||
3 | 100.00 | 6.02677 | 3.36065 | 100.00 | 6.45903 | 3.60106 | 0.21087 | 0.46892 | ||||
97.76 | 0.69453 | 99.30 | 0.74309 | |||||||||
30 | – | – | – | 98.12 | 5.07645 | 2.84264 | 98.12 | 6.39162 | 3.51761 | 0.21496 | 0.47672 | |
96.88 | 0.60883 | 98.68 | 0.64361 | |||||||||
60 | 36 | 24 | 0.75 | 1 | 99.98 | 5.03231 | 2.82001 | 99.98 | 6.77235 | 3.70549 | 0.19640 | 0.43171 |
97.94 | 0.60770 | 98.92 | 0.63864 | |||||||||
2 | 99.96 | 4.99889 | 2.76598 | 100.00 | 6.78508 | 3.66938 | 0.19302 | 0.38688 | ||||
97.08 | 0.53307 | 98.72 | 0.55368 | |||||||||
3 | 100.00 | 5.07471 | 2.79803 | 100.00 | 6.82111 | 3.68090 | 0.19267 | 0.39424 | ||||
97.20 | 0.52134 | 98.34 | 0.54069 | |||||||||
1.25 | 1 | 99.96 | 4.37391 | 2.47518 | 99.96 | 5.74542 | 3.17454 | 0.20554 | 0.46446 | |||
97.20 | 0.57646 | 98.78 | 0.60367 | |||||||||
2 | 99.94 | 4.41337 | 2.46448 | 99.94 | 5.83758 | 3.18608 | 0.19541 | 0.41668 | ||||
96.76 | 0.51559 | 98.80 | 0.53458 | |||||||||
3 | 99.96 | 4.44709 | 2.47536 | 99.96 | 5.88787 | 3.20458 | 0.19678 | 0.42178 | ||||
96.98 | 0.50363 | 98.66 | 0.52129 | |||||||||
48 | 24 | 0.75 | 1 | 100.00 | 4.87704 | 2.70037 | 100.00 | 6.38144 | 3.46231 | 0.19400 | 0.42938 | |
98.38 | 0.52369 | 99.08 | 0.54318 | |||||||||
2 | 100.00 | 4.87786 | 2.68734 | 100.00 | 6.38594 | 3.44970 | 0.19267 | 0.40875 | ||||
97.52 | 0.49682 | 98.68 | 0.51346 | |||||||||
3 | 100.00 | 4.83923 | 2.67338 | 100.00 | 6.36455 | 3.44493 | 0.19513 | 0.43037 | ||||
97.94 | 0.50752 | 99.04 | 0.52531 | |||||||||
1.25 | 1 | 99.52 | 4.1727 | 2.3373 | 99.52 | 5.53689 | 3.02835 | 0.20409 | 0.46233 | |||
96.84 | 0.50191 | 98.42 | 0.51981 | |||||||||
2 | 99.58 | 4.16144 | 2.31843 | 99.58 | 5.53870 | 3.01455 | 0.19897 | 0.43895 | ||||
97.06 | 0.47543 | 98.34 | 0.49040 | |||||||||
3 | 99.54 | 4.19359 | 2.34165 | 99.54 | 5.55192 | 3.02897 | 0.20262 | 0.46293 | ||||
97.38 | 0.48971 | 98.50 | 0.50602 | |||||||||
60 | – | – | – | 94.96 | 3.73590 | 2.08512 | 94.96 | 5.82791 | 3.13701 | 0.21386 | 0.47246 | |
96.74 | 0.43435 | 98.22 | 0.44610 |
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Abdel-Hamid, A.H.; Hashem, A.F. Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function. Mathematics 2021, 9, 1510. https://doi.org/10.3390/math9131510
Abdel-Hamid AH, Hashem AF. Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function. Mathematics. 2021; 9(13):1510. https://doi.org/10.3390/math9131510
Chicago/Turabian StyleAbdel-Hamid, Alaa H., and Atef F. Hashem. 2021. "Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function" Mathematics 9, no. 13: 1510. https://doi.org/10.3390/math9131510
APA StyleAbdel-Hamid, A. H., & Hashem, A. F. (2021). Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function. Mathematics, 9(13), 1510. https://doi.org/10.3390/math9131510