Exact Likelihood Inference for an Exponential Parameter under Generalized Adaptive Progressive Hybrid Censoring
Abstract
:1. Introduction
2. Model Description and Inference
2.1. Model Description
2.2. MLE
2.3. Exact Inference for MLE
- (a)
- The conditional joint density (ConJD) of given , iswhere for .
- (b)
- For and , the ConJD of given and , is
- (c)
- For , the ConJD of given , is
3. Real Data Analysis and Simulation Results
3.1. Real Data Analysis
3.2. Simulation Results
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Proof of Theorem 1
Appendix B. Proof of Theorem 2
Appendix C. Proof of Theorem 3
References
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i | 1 | 2 | 3 | 4 |
0.18999 | 0.77997 | 0.95993 | 1.30996 | |
0 | 0 | 3 | 0 | |
i | 5 | 6 | 7 | 8 |
2.77986 | 4.84962 | 6.49999 | 7.35000 | |
3 | 0 | 0 | 5 |
C.I for | |||||
---|---|---|---|---|---|
Case | MSE | SE. | |||
Case I | 9.08855 | 14.9261 | 3.6217 | (4.99597, 20.96100) | (5.48302, 18.17892) |
Case II | 10.80235 | 23.2914 | 4.5846 | (5.98840, 25.00965) | (6.56902, 21.69869) |
Case III | 11.89555 | 45.7945 | 6.6352 | (9.88020, 41.89090) | (10.85072, 36.28145) |
MSE (Bias) | |||||||
---|---|---|---|---|---|---|---|
20 | 0.8 | 18 | (17*0,2) † | 0.0807 (0.0353) | 0.0725 (0.0228) | 0.0650 (0.0154) | 0.0613 (0.0109) |
(2,17*0) ‡ | 0.0913 (0.0399) | 0.0773 (0.0274) | 0.0736 (0.0200) | 0.0667 (0.0165) | |||
(1,16*0,1) | 0.0900 (0.4210) | 0.0751 (0.0242) | 0.0669 (0.0176) | 0.0641 (0.0145) | |||
16 | (15*0,4) | 0.0914 (0.0286) | 0.0779 (0.0165) | 0.0698 (0.0081) | 0.0681 (0.0070) | ||
(4,15*0) | 0.1129 (0.0379) | 0.0967 (0.2810) | 0.0812 (0.0198) | 0.0782 (0.0185) | |||
(2,14*0,2) | 0.1005 (0.0355) | 0.0841 (0.0231) | 0.0774 (0.0168) | 0.0710 (0.0101) | |||
14 | (13*0,6) | 0.0863 (0.0188) | 0.0765 (0.0101) | 0.0754 (0.0094) | 0.0752 (0.0094) | ||
(6,13*0) | 0.1288 (0.0522) | 0.1077 (0.0377) | 0.0914 (0.0267) | 0.0852 (0.0207) | |||
(3,12*0,3) | 0.1101 (0.0420) | 0.0857 (0.0191) | 0.0790 (0.0123) | 0.0769 (0.0100) | |||
12 | (11*0,8) | 0.0856 (0.0133) | 0.0826 (0.0115) | 0.0826 (0.0115) | 0.0826 (0.0115) | ||
(8,11*0) | 0.1478 (0.0542) | 0.1137 (0.0319) | 0.1002 (0.0270) | 0.0937 (0.0236) | |||
(4,10*0,4) | 0.1051 (0.0311) | 0.0897 (0.0171) | 0.0839 (0.0121) | 0.0830 (0.0117) | |||
30 | 0.8 | 26 | (25*0,4) | 0.0502 (0.0168) | 0.0431 (0.0075) | 0.0408 (0.0023) | 0.0395 (0.0007) |
(4,55*0) | 0.0613 (0.0256) | 0.0498 (0.0140) | 0.0451 (0.0090) | 0.0426 (0.0059) | |||
(2,24*0,2) | 0.0551 (0.0223) | 0.0464 (0.0129) | 0.0428 (0.0067) | 0.0408 (0.0030) | |||
24 | (23*0,6) | 0.0503 (0.0140) | 0.0446 (0.0050) | 0.0426 (0.0022) | 0.0423 (0.0020) | ||
(6,23*0) | 0.0658 (0.0247) | 0.0531 (0.0123) | 0.0490 (0.0081) | 0.0463 (0.0061) | |||
(3,22*0,3) | 0.0553 (0.0194) | 0.0483 (0.0094) | 0.0449 (0.0057) | 0.0429 (0.0029) | |||
22 | (21*0,8) | 0.0498 (0.0115) | 0.0444 (0.0040) | 0.0438 (0.0036) | 0.0437 (0.0036) | ||
(8,21*0) | 0.0682 (0.0326) | 0.0569 (0.0222) | 0.0530 (0.0167) | 0.0488 (0.0116) | |||
(4,20*0,4) | 0.0554 (0.0245) | 0.0491 (0.0120) | 0.0456 (0.0055) | 0.0442 (0.0039) | |||
20 | (19*0,10) | 0.0548 (0.0060) | 0.0512 (0.0036) | 0.0511 (0.0036) | 0.0511 (0.0036) | ||
(10,19*0) | 0.0824 (0.0344) | 0.0684 (0.0214) | 0.0606 (0.0128) | 0.0572 (0.0108) | |||
(5,18*0,5) | 0.0649 (0.0205) | 0.0558 (0.0077) | 0.0522 (0.0042) | 0.0513 (0.0037) | |||
40 | 0.8 | 36 | (35*0,4) | 0.0345 (0.0124) | 0.0309 (0.0056) | 0.0278 (−0.0007) | 0.0269 (−0.0031) |
(4,35*0) | 0.0371 (0.0149) | 0.0340 (0.0085) | 0.0317 (0.0057) | 0.0294 (0.0014) | |||
(2,34*0,2) | 0.0359 (0.0122) | 0.0325 (0.0085) | 0.0295 (0.0025) | 0.0279 (−0.0006) | |||
32 | (27*0,8) | 0.0368 (0.0073) | 0.0341 (0.0000) | 0.0323 (−0.0023) | 0.0323 (−0.0023) | ||
(8,27*0) | 0.0438 (0.0132) | 0.0389 (0.0095) | 0.0360 (0.0051) | 0.0349 (0.0033) | |||
(4,26*0,4) | 0.0392 (0.0121) | 0.0348 (0.0038) | 0.0343 (0.0007) | 0.0331 (−0.0013) | |||
28 | (35*0,12) | 0.0382 (0.0027) | 0.0358 (−0.0002) | 0.0356 (−0.0003) | 0.0356 (−0.0003) | ||
(12,35*0) | 0.0511 (0.0165) | 0.0436 (0.0099) | 0.0420 (0.0088) | 0.0401 (0.0069) | |||
(6,34*0,6) | 0.0435 (0.0129) | 0.0388 (0.0042) | 0.0363 (0.0004) | 0.0356 (−0.0003) | |||
24 | (27*0,16) | 0.0424 (0.0021) | 0.0423 (0.0020) | 0.0423 (0.0020) | 0.0423 (0.0020) | ||
(16,27*0) | 0.0650 (0.0247) | 0.0525 (0.0119) | 0.0488 (0.0079) | 0.0462 (0.0059) | |||
(8,26*0,8) | 0.0485 (0.0102) | 0.0430 (0.0026) | 0.0424 (0.0021) | 0.0423 (0.0020) | |||
50 | 0.8 | 46 | (45*0,4) | 0.0265 (0.0070) | 0.0235 (0.0031) | 0.0216 (−0.0010) | 0.0208 (−0.0030) |
(4,45*0) | 0.0293 (0.0085) | 0.0258 (0.0049) | 0.0236 (0.0011) | 0.0225 (−0.0005) | |||
(2,44*0,2) | 0.0275 (0.0069) | 0.0243 (0.0040) | 0.0224 (0.0003) | 0.0215 (−0.0015) | |||
42 | (41*0,8) | 0.0280 (0.0082) | 0.0256 (0.0017) | 0.0234 (−0.0025) | 0.0234 (−0.0026) | ||
(8,41*0) | 0.0324 (0.0094) | 0.0288 (0.0042) | 0.0272 (0.0033) | 0.0263 (0.0030) | |||
(4,40*0,4) | 0.0301 (0.0087) | 0.0265 (0.0043) | 0.0254 (0.0016) | 0.0240 (−0.0015) | |||
38 | (37*0,12) | 0.0305 (0.0021) | 0.0275 (−0.0033) | 0.0272 (−0.0035) | 0.0272 (−0.0035) | ||
(12,37*0) | 0.0410 (0.0144) | 0.0345 (0.0063) | 0.0312 (0.0015) | 0.0296 (0.0004) | |||
(6,36*0,6) | 0.0340 (0.0089) | 0.0298 (0.0017) | 0.0282 (−0.0023) | 0.0273 (−0.0035) | |||
34 | (33*0,16) | 0.0295 (−0.0019) | 0.0286 (−0.0029) | 0.0286 (−0.0029) | 0.0286 (−0.0029) | ||
(16,33*0) | 0.0396 (0.0048) | 0.0341 (0.0012) | 0.0323 (0.0007) | 0.0311 (0.0000) | |||
(8,32*0,8) | 0.0326 (0.0036) | 0.0292 (−0.0024) | 0.0287 (−0.0030) | 0.0287 (−0.0030) | |||
60 | 0.8 | 54 | (53*0,6) | 0.0217 (0.0041) | 0.0191 (−0.0001) | 0.0180 (−0.0048) | 0.0177 (−0.0058) |
(6,53*0) | 0.0249 (0.0069) | 0.0210 (0.0013) | 0.0194 (−0.0018) | 0.0188 (−0.0026) | |||
(3,52*0,3) | 0.0228 (0.0052) | 0.0198 (0.0008) | 0.0187 (−0.0017) | 0.0180 (−0.0051) | |||
48 | (47*0,12) | 0.0231 (0.0016) | 0.0213 (−0.0035) | 0.0207 (−0.0044) | 0.0207 (−0.0044) | ||
(12,47*0) | 0.0296 (0.0088) | 0.0246 (0.0027) | 0.0232 (−0.0011) | 0.0219 (−0.0030) | |||
(6,46*0,6) | 0.0245 (0.0048) | 0.0225 (−0.0003) | 0.0216 (−0.0026) | 0.0210 (−0.0040) | |||
42 | (41*0,18) | 0.0244 (−0.0011) | 0.0234 (−0.0026) | 0.0234 (−0.0026) | 0.0234 (−0.0026) | ||
(18,41*0) | 0.0323 (0.0094) | 0.0288 (0.0043) | 0.0271 (0.0034) | 0.0263 (0.0030) | |||
(9,40*0,9) | 0.0276 (0.0077) | 0.0250 (0.0002) | 0.0235 (−0.0025) | 0.0234 (−0.0026) | |||
36 | (35*0,24) | 0.0267 (−0.0035) | 0.0267 (−0.0035) | 0.0267 (−0.0035) | 0.0267 (−0.0035) | ||
(24,35*0) | 0.0368 (0.0145) | 0.0337 (0.0085) | 0.0317 (0.0056) | 0.0295 (0.0013) | |||
(12,34*0,12) | 0.0297 (0.0020) | 0.0268 (−0.0034) | 0.0267 (−0.0035) | 0.0267 (−0.0035) |
Coverage Percentage (Confidence Length) | |||||||
---|---|---|---|---|---|---|---|
20 | 0.8 | 18 | (17*0,2) † | 94.8 (1.0548) | 94.5 (0.9846) | 94.4 (0.9525) | 94.4 (0.9385) |
(2,17*0) ‡ | 93.8 (1.1183) | 94.0 (1.0371) | 94.3 (0.9945) | 94.4 (0.9709) | |||
(1,16*0,1) | 94.3 (1.0925) | 94.8 (1.0069) | 94.4 (0.9688) | 94.4 (0.9509) | |||
16 | (15*0,4) | 93.2 (1.0608) | 93.1 (1.0083) | 93.1 (0.9894) | 93.1 (0.9869) | ||
(4,15*0) | 93.8 (1.1865) | 94.0 (1.1043) | 93.5 (1.0559) | 93.2 (1.0340) | |||
(2,14*0,2) | 93.5 (1.1170) | 93.1 (1.0427) | 93.1 (1.0112) | 93.1 (0.9938) | |||
14 | (13*0,6) | 93.5 (1.0837) | 93.4 (1.0595) | 93.3 (1.0577) | 93.3 (1.0575) | ||
(6,13*0) | 93.5 (1.2875) | 93.1 (1.1916) | 93.4 (1.1359) | 93.4 (1.1056) | |||
(3,12*0,3) | 93.5 (1.1650) | 93.5 (1.0862) | 93.5 (1.0647) | 93.3 (1.0591) | |||
12 | (11*0,8) | 93.8 (1.1490) | 93.8 (1.1446) | 93.8 (1.1446) | 93.8 (1.1446) | ||
(8,11*0) | 93.8 (1.3988) | 94.2 (1.2792) | 94.0 (1.2301) | 94.0 (1.2006) | |||
(4,10*0,4) | 93.8 (1.2077) | 93.8 (1.1580) | 93.8 (1.1460) | 93.8 (1.4490) | |||
30 | 0.8 | 26 | (25*0,4) | 94.3 (0.8377) | 94.8 (0.7898) | 94.8 (0.7737) | 94.6 (0.7699) |
(4,55*0) | 94.6 (0.9104) | 94.0 (0.8461) | 94.7 (0.8145) | 94.8 (0.7965) | |||
(2,24*0,2) | 93.9 (0.8728) | 94.0 (0.8157) | 93.8 (0.7879) | 93.8 (0.7757) | |||
24 | (23*0,6) | 93.6 (0.8418) | 93.4 (0.8086) | 93.5 (0.8023) | 93.3 (0.8018) | ||
(6,23*0) | 94.1 (0.9456) | 94.1 (0.8784) | 94.1 (0.8470) | 93.8 (0.8295) | |||
(3,22*0,3) | 94.1 (0.8855) | 93.6 (0.8313) | 93.4 (0.8116) | 93.4 (0.8038) | |||
22 | (21*0,8) | 93.4 (0.8574) | 93.3 (0.8398) | 93.2 (0.8388) | 93.2 (0.8387) | ||
(8,21*0) | 93.8 (0.9978) | 93.1 (0.9292) | 93.9 (0.8936) | 94.4 (0.8713) | |||
(4,20*0,4) | 93.4 (0.9100) | 93.5 (0.8589) | 93.3 (0.8425) | 93.3 (0.8394) | |||
20 | (19*0,10) | 93.9 (0.8855) | 93.9 (0.8799) | 93.9 (0.8797) | 93.9 (0.8797) | ||
(10,19*0) | 93.4 (1.0503) | 93.2 (0.9745) | 93.1 (0.9331) | 93.0 (0.9138) | |||
(5,18*0,5) | 93.0 (0.9320) | 93.8 (0.8898) | 93.9 (0.8811) | 93.9 (0.8799) | |||
40 | 0.8 | 36 | (35*0,4) | 94.4 (0.7193) | 94.4 (0.6765) | 94.4 (0.6573) | 94.3 (0.6520) |
(4,35*0) | 94.5 (0.7607) | 94.3 (0.7134) | 94.5 (0.6890) | 94.1 (0.6726) | |||
(2,34*0,2) | 94.3 (0.7376) | 94.2 (0.6947) | 94.3 (0.6694) | 94.3 (0.6580) | |||
32 | (27*0,8) | 94.2 (0.7208) | 94.1 (0.6956) | 94.1 (0.6914) | 94.1 (0.6914) | ||
(8,27*0) | 94.7 (0.8049) | 94.5 (0.7580) | 94.3 (0.7305) | 94.6 (0.7158) | |||
(4,26*0,4) | 94.5 (0.7583) | 94.4 (0.7134) | 94.1 (0.6979) | 94.9 (0.6929) | |||
28 | (35*0,12) | 94.7 (0.7466) | 94.5 (0.7408) | 94.5 (0.7406) | 94.5 (0.7406) | ||
(12,35*0) | 94.0 (0.8650) | 94.6 (0.8108) | 94.9 (0.7851) | 94.5 (0.7688) | |||
(6,34*0,6) | 94.7 (0.7854) | 94.7 (0.7501) | 94.5 (0.7418) | 94.5 (0.7406) | |||
24 | (27*0,16) | 93.3 (0.8020) | 93.3 (0.8017) | 93.3 (0.8017) | 93.3 (0.8017) | ||
(16,27*0) | 93.1 (0.9444) | 93.1 (0.8773) | 93.1 (0.8464) | 93.8 (0.8291) | |||
(8,26*0,8) | 93.6 (0.8227) | 93.4 (0.8031) | 93.3 (0.8019) | 93.3 (0.8017) | |||
50 | 0.8 | 46 | (45*0,4) | 94.2 (0.6377) | 94.4 (0.6011) | 94.8 (0.5834) | 94.8 (0.5773) |
(4,45*0) | 94.0 (0.6664) | 94.5 (0.6278) | 94.5 (0.6056) | 94.5 (0.5937) | |||
(2,44*0,2) | 94.9 (0.6507) | 95.0 (0.6137) | 94.7 (0.5927) | 94.7 (0.5827) | |||
42 | (41*0,8) | 94.8 (0.6403) | 94.8 (0.6107) | 94.7 (0.6035) | 94.8 (0.6033) | ||
(8,41*0) | 94.5 (0.6984) | 94.1 (0.6565) | 94.9 (0.6360) | 94.1 (0.6245) | |||
(4,40*0,4) | 94.8 (0.6668) | 94.8 (0.6284) | 94.9 (0.6117) | 94.8 (0.6050) | |||
38 | (37*0,12) | 94.6 (0.6461) | 94.8 (0.6340) | 94.8 (0.6337) | 94.8 (0.6337) | ||
(12,37*0) | 94.9 (0.7398) | 94.9 (0.6919) | 94.5 (0.6666) | 94.4 (0.6541) | |||
(6,36*0,6) | 94.8 (0.6824) | 94.7 (0.6462) | 94.6 (0.6356) | 94.8 (0.6337) | |||
34 | (33*0,16) | 94.7 (0.6719) | 94.7 (0.6703) | 94.7 (0.6703) | 94.7 (0.6703) | ||
(16,33*0) | 94.4 (0.7715) | 94.3 (0.7272) | 94.6 (0.7050) | 94.2 (0.6920) | |||
(8,32*0,8) | 94.7 (0.6880) | 94.7 (0.6712) | 94.7 (0.6702) | 94.7 (0.6702) | |||
60 | 0.8 | 54 | (53*0,6) | 94.8 (0.5796) | 94.9 (0.5469) | 94.3 (0.5332) | 94.2 (0.5306) |
(6,53*0) | 94.4 (0.6136) | 94.9 (0.5763) | 94.9 (0.5567) | 94.5 (0.5465) | |||
(3,52*0,3) | 95.2 (0.5956) | 95.2 (0.5606) | 94.9 (0.5429) | 94.2 (0.5337) | |||
48 | (47*0,12) | 94.1 (0.5814) | 94.2 (0.5646) | 94.2 (0.5633) | 94.2 (0.5633) | ||
(12,47*0) | 94.5 (0.6523) | 94.1 (0.6123) | 94.9 (0.5909) | 94.5 (0.5792) | |||
(6,46*0,6) | 94.3 (0.6114) | 94.0 (0.5776) | 94.1 (0.5664) | 94.2 (0.5637) | |||
42 | (41*0,18) | 94.9 (0.6055) | 94.8 (0.6033) | 94.8 (0.6033) | 94.8 (0.6033) | ||
(18,41*0) | 94.5 (0.6981) | 94.1 (0.6564) | 94.9 (0.6359) | 94.1 (0.6244) | |||
(9,40*0,9) | 94.7 (0.6343) | 94.9 (0.6078) | 94.8 (0.6034) | 94.8 (0.6033) | |||
36 | (35*0,24) | 94.3 (0.6510) | 94.3 (0.6510) | 94.3 (0.6510) | 94.3 (0.6510) | ||
(24,35*0) | 94.5 (0.7592) | 94.3 (0.7129) | 94.4 (0.6886) | 94.0 (0.6725) | |||
(12,34*0,12) | 94.3 (0.6621) | 94.3 (0.6513) | 94.3 (0.6510) | 94.3 (0.6510) |
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Lee, H.; Lee, K. Exact Likelihood Inference for an Exponential Parameter under Generalized Adaptive Progressive Hybrid Censoring. Symmetry 2020, 12, 1149. https://doi.org/10.3390/sym12071149
Lee H, Lee K. Exact Likelihood Inference for an Exponential Parameter under Generalized Adaptive Progressive Hybrid Censoring. Symmetry. 2020; 12(7):1149. https://doi.org/10.3390/sym12071149
Chicago/Turabian StyleLee, Hyojin, and Kyeongjun Lee. 2020. "Exact Likelihood Inference for an Exponential Parameter under Generalized Adaptive Progressive Hybrid Censoring" Symmetry 12, no. 7: 1149. https://doi.org/10.3390/sym12071149