Research Article
Modified One-Parameter Liu Estimator for the Linear Regression Model
| | | 0.7 | 0.8 | 0.9 | 0.99 | | OLS | Ridge | Liu | MLiu | OLS | Ridge | Liu | MLiu | OLS | Ridge | Liu | MLiu | OLS | Ridge | Liu | MLiu |
| 1 | 0.1 | 0.0608 | 0.0607 | 0.0598 | 0.0595 | 0.0856 | 0.0854 | 0.0835 | 0.0830 | 0.1620 | 0.1611 | 0.1543 | 0.1527 | 1.5604 | 1.4782 | 1.0328 | 0.9358 | 0.2 | 0.0608 | 0.0606 | 0.0599 | 0.0594 | 0.0856 | 0.0851 | 0.0837 | 0.0828 | 0.1620 | 0.1602 | 0.1552 | 0.1519 | 1.5604 | 1.4031 | 1.0840 | 0.8901 | 0.3 | 0.0608 | 0.0605 | 0.0600 | 0.0593 | 0.0856 | 0.0849 | 0.0839 | 0.0825 | 0.1620 | 0.1593 | 0.1560 | 0.1511 | 1.5604 | 1.3343 | 1.1371 | 0.8463 | 0.4 | 0.0608 | 0.0603 | 0.0601 | 0.0592 | 0.0856 | 0.0846 | 0.0842 | 0.0823 | 0.1620 | 0.1585 | 0.1568 | 0.1504 | 1.5604 | 1.2712 | 1.1921 | 0.8042 | 0.5 | 0.0608 | 0.0602 | 0.0602 | 0.0591 | 0.0856 | 0.0844 | 0.0844 | 0.0821 | 0.1620 | 0.1576 | 0.1577 | 0.1496 | 1.5604 | 1.2132 | 1.2488 | 0.7641 | 0.6 | 0.0608 | 0.0601 | 0.0603 | 0.0590 | 0.0856 | 0.0841 | 0.0846 | 0.0819 | 0.1620 | 0.1568 | 0.1585 | 0.1488 | 1.5604 | 1.1598 | 1.3075 | 0.7257 | 0.7 | 0.0608 | 0.0600 | 0.0605 | 0.0589 | 0.0856 | 0.0839 | 0.0849 | 0.0817 | 0.1620 | 0.1559 | 0.1594 | 0.1481 | 1.5604 | 1.1105 | 1.3679 | 0.6892 | 0.8 | 0.0608 | 0.0599 | 0.0606 | 0.0588 | 0.0856 | 0.0837 | 0.0851 | 0.0815 | 0.1620 | 0.1551 | 0.1602 | 0.1473 | 1.5604 | 1.0649 | 1.4302 | 0.6546 | 0.9 | 0.0608 | 0.0598 | 0.0607 | 0.0587 | 0.0856 | 0.0834 | 0.0854 | 0.0812 | 0.1620 | 0.1543 | 0.1611 | 0.1466 | 1.5604 | 1.0226 | 1.4944 | 0.6218 | 1.0 | 0.0608 | 0.0596 | 0.0608 | 0.0586 | 0.0856 | 0.0832 | 0.0856 | 0.0810 | 0.1620 | 0.1535 | 0.1620 | 0.1459 | 1.5604 | 0.9834 | 1.5604 | 0.5908 |
| 5 | 0.1 | 1.5209 | 1.5177 | 1.4931 | 1.4869 | 2.1402 | 2.1338 | 2.0844 | 2.0722 | 4.0504 | 4.0274 | 3.8518 | 3.8085 | 39.0100 | 36.9446 | 25.4493 | 22.8472 | 0.2 | 1.5209 | 1.5146 | 1.4961 | 1.4839 | 2.1402 | 2.1275 | 2.0906 | 2.0660 | 4.0504 | 4.0047 | 3.8736 | 3.7869 | 39.0100 | 35.0465 | 26.8065 | 21.6023 | 0.3 | 1.5209 | 1.5115 | 1.4992 | 1.4808 | 2.1402 | 2.1213 | 2.0967 | 2.0599 | 4.0504 | 3.9822 | 3.8954 | 3.7654 | 39.0100 | 33.2972 | 28.2010 | 20.3948 | 0.4 | 1.5209 | 1.5084 | 1.5023 | 1.4778 | 2.1402 | 2.1150 | 2.1029 | 2.0539 | 4.0504 | 3.9598 | 3.9174 | 3.7440 | 39.0100 | 31.6813 | 29.6329 | 19.2247 | 0.5 | 1.5209 | 1.5053 | 1.5054 | 1.4747 | 2.1402 | 2.1088 | 2.1091 | 2.0478 | 4.0504 | 3.9377 | 3.9393 | 3.7226 | 39.0100 | 30.1850 | 31.1023 | 18.0919 | 0.6 | 1.5209 | 1.5022 | 1.5085 | 1.4717 | 2.1402 | 2.1027 | 2.1153 | 2.0417 | 4.0504 | 3.9158 | 3.9614 | 3.7013 | 39.0100 | 28.7964 | 32.6090 | 16.9966 | 0.7 | 1.5209 | 1.4991 | 1.5116 | 1.4686 | 2.1402 | 2.0965 | 2.1215 | 2.0357 | 4.0504 | 3.8941 | 3.9835 | 3.6801 | 39.0100 | 27.5052 | 34.1531 | 15.9387 | 0.8 | 1.5209 | 1.4961 | 1.5147 | 1.4656 | 2.1402 | 2.0904 | 2.1277 | 2.0296 | 4.0504 | 3.8726 | 4.0057 | 3.6590 | 39.0100 | 26.3022 | 35.7347 | 14.9181 | 0.9 | 1.5209 | 1.4930 | 1.5178 | 1.4626 | 2.1402 | 2.0843 | 2.1339 | 2.0236 | 4.0504 | 3.8512 | 4.0280 | 3.6379 | 39.0100 | 25.1794 | 37.3536 | 13.9350 | 1.0 | 1.5209 | 1.4900 | 1.5209 | 1.4595 | 2.1402 | 2.0783 | 2.1402 | 2.0175 | 4.0504 | 3.8301 | 4.0504 | 3.6169 | 39.0100 | 24.1296 | 39.0100 | 12.9893 |
| 10 | 0.1 | 6.0835 | 6.0709 | 5.9721 | 5.9475 | 8.5606 | 8.5353 | 8.3373 | 8.2882 | 16.2015 | 16.1096 | 15.4061 | 15.2323 | 156.0399 | 147.7764 | 101.7462 | 91.3151 | 0.2 | 6.0835 | 6.0584 | 5.9844 | 5.9352 | 8.5606 | 8.5101 | 8.3620 | 8.2636 | 16.2015 | 16.0186 | 15.4934 | 15.1457 | 156.0399 | 140.1802 | 107.1845 | 86.3224 | 0.3 | 6.0835 | 6.0459 | 5.9967 | 5.9230 | 8.5606 | 8.4850 | 8.3867 | 8.2392 | 16.2015 | 15.9284 | 15.5810 | 15.0595 | 156.0399 | 133.1789 | 112.7715 | 81.4783 | 0.4 | 6.0835 | 6.0335 | 6.0091 | 5.9107 | 8.5606 | 8.4600 | 8.4114 | 8.2147 | 16.2015 | 15.8390 | 15.6688 | 14.9735 | 156.0399 | 126.7096 | 118.5070 | 76.7827 | 0.5 | 6.0835 | 6.0211 | 6.0214 | 5.8985 | 8.5606 | 8.4352 | 8.4362 | 8.1903 | 16.2015 | 15.7504 | 15.7569 | 14.8878 | 156.0399 | 120.7180 | 124.3910 | 72.2358 | 0.6 | 6.0835 | 6.0088 | 6.0338 | 5.8863 | 8.5606 | 8.4105 | 8.4610 | 8.1659 | 16.2015 | 15.6626 | 15.8453 | 14.8024 | 156.0399 | 115.1568 | 130.4237 | 67.8373 | 0.7 | 6.0835 | 5.9965 | 6.0462 | 5.8741 | 8.5606 | 8.3859 | 8.4858 | 8.1416 | 16.2015 | 15.5756 | 15.9339 | 14.7172 | 156.0399 | 109.9844 | 136.6049 | 63.5875 | 0.8 | 6.0835 | 5.9842 | 6.0586 | 5.8619 | 8.5606 | 8.3614 | 8.5107 | 8.1173 | 16.2015 | 15.4893 | 16.0229 | 14.6323 | 156.0399 | 105.1643 | 142.9346 | 59.4862 | 0.9 | 6.0835 | 5.9720 | 6.0710 | 5.8497 | 8.5606 | 8.3370 | 8.5356 | 8.0931 | 16.2015 | 15.4038 | 16.1120 | 14.5476 | 156.0399 | 100.6645 | 149.4130 | 55.5334 | 1.0 | 6.0835 | 5.9598 | 6.0835 | 5.8375 | 8.5606 | 8.3127 | 8.5606 | 8.0689 | 16.2015 | 15.3190 | 16.2015 | 14.4633 | 156.0399 | 96.4563 | 156.0399 | 51.7293 |
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