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 | 0.8375 | 0.8019 | 0.6073 | 0.5645 | 1.3038 | 1.2137 | 0.8030 | 0.7166 | 2.7626 | 2.3689 | 1.1868 | 0.9622 | 30.134 | 10.720 | 2.2263 | 0.9170 | 0.2 | 0.8375 | 0.7694 | 0.6299 | 0.5442 | 1.3038 | 1.1356 | 0.8496 | 0.6767 | 2.7626 | 2.0758 | 1.3162 | 0.8670 | 30.134 | 6.4830 | 3.5481 | 0.9295 | 0.3 | 0.8375 | 0.7395 | 0.6532 | 0.5246 | 1.3038 | 1.0673 | 0.8985 | 0.6392 | 2.7626 | 1.8492 | 1..4571 | 0.7833 | 30.134 | 4.5630 | 5.3147 | 1.3867 | 0.4 | 0.8375 | 0.7121 | 0.6773 | 0.5058 | 1.3038 | 1.0072 | 0.9496 | 0.6039 | 2.7626 | 1.6691 | 1.6093 | 0.7109 | 30.134 | 3.4720 | 7.5260 | 2.2886 | 0.5 | 0.8375 | 0.6868 | 0.7021 | 0.4877 | 1.3038 | 0.9539 | 1.0030 | 0.5708 | 2.7626 | 1.5224 | 1.7730 | 0.6500 | 30.134 | 2.7762 | 10.182 | 3.6354 | 0.6 | 0.8375 | 0.6635 | 0.7277 | 0.4705 | 1.3038 | 0.9063 | 1.0586 | 0.5400 | 2.7626 | 1.4006 | 1.9481 | 0.6005 | 30.134 | 2.2996 | 13.283 | 5.4268 | 0.7 | 0.8375 | 0.6418 | 0.7540 | 0.4539 | 1.3038 | 0.8636 | 1.1166 | 0.5115 | 2.7626 | 1.2980 | 2.1346 | 0.5624 | 30.134 | 1.9565 | 16.828 | 7.6631 | 0.8 | 0.8375 | 0.6217 | 0.7811 | 0.4381 | 1.3038 | 0.8252 | 1.1767 | 0.4853 | 2.7626 | 1.2104 | 2.3325 | 0.5357 | 30.134 | 1.7004 | 20.819 | 10.344 | 0.9 | 0.8375 | 0.6030 | 0.8089 | 0.4231 | 1.3038 | 0.7903 | 1.2391 | 0.4613 | 2.7626 | 1.1348 | 2.5418 | 0.5204 | 30.134 | 1.5038 | 25.254 | 13.470 | 1.0 | 0.8375 | 0.5855 | 0.8375 | 0.4088 | 1.3038 | 0.7587 | 1.3038 | 0.4395 | 2.7626 | 1.0688 | 2.7626 | 0.5166 | 30.134 | 1.3493 | 30.134 | 17.040 |
| 5 | 0.1 | 20.936 | 20.040 | 15.024 | 13.885 | 32.596 | 30.329 | 19.803 | 17.525 | 69.065 | 59.182 | 29.022 | 23.128 | 753.34 | 266.98 | 50.691 | 15.905 | 0.2 | 20.936 | 19.216 | 15.618 | 13.339 | 32.596 | 28.357 | 21.019 | 16.463 | 69.065 | 51.794 | 32.378 | 20.591 | 753.34 | 160.14 | 84.633 | 15.061 | 0.3 | 20.936 | 18.457 | 16.227 | 12.809 | 32.596 | 26.624 | 22.286 | 15.452 | 69.065 | 46.060 | 36.008 | 18.328 | 753.34 | 111.36 | 129.61 | 25.249 | 0.4 | 20.936 | 17.755 | 16.852 | 12.295 | 32.596 | 25.091 | 23.605 | 14.493 | 69.065 | 41.478 | 39.911 | 16.337 | 753.34 | 83.415 | 185.61 | 46.470 | 0.5 | 20..936 | 17.104 | 17.493 | 11.797 | 32.596 | 23.724 | 24.975 | 13.586 | 69.065 | 37.727 | 44.087 | 14.620 | 753.34 | 65.445 | 252.65 | 78.724 | 0.6 | 20.936 | 16.498 | 18.150 | 11.315 | 32.596 | 22.498 | 26.396 | 12.729 | 69.065 | 34.597 | 48.536 | 14.176 | 753.34 | 53.028 | 330.73 | 122.01 | 0.7 | 20.936 | 15.933 | 18.823 | 10.848 | 32.596 | 21.392 | 27.869 | 11.924 | 69.065 | 31.943 | 53.258 | 12.005 | 753.34 | 44.012 | 419.83 | 176.33 | 0.8 | 20.936 | 15.405 | 19.512 | 10.398 | 32.596 | 20.389 | 29.393 | 11.171 | 69.065 | 29.662 | 58.254 | 11.107 | 753.34 | 37.220 | 519.97 | 241.68 | 0.9 | 20.936 | 14.911 | 20.216 | 9.9630 | 32.596 | 19.474 | 30.969 | 10.469 | 69.065 | 27.679 | 63.523 | 10.482 | 753.34 | 31.956 | 631.14 | 318.06 | 1.0 | 20.936 | 14.447 | 20.936 | 9.5441 | 32.596 | 18.638 | 32.596 | 9.8179 | 69.065 | 25.938 | 69.065 | 10.131 | 753.34 | 27.782 | 753.34 | 405.48 |
| 10 | 0.1 | 83.746 | 80.157 | 60.064 | 55.497 | 130.38 | 121.31 | 79.156 | 70.028 | 276.26 | 236.71 | 115.97 | 92.358 | 3013.4 | 1067.6 | 201.97 | 62.631 | 0.2 | 83.746 | 76.859 | 62.443 | 53.307 | 130.38 | 113.42 | 84.028 | 65.771 | 276.26 | 207.15 | 129.41 | 82.190 | 3013.4 | 640.15 | 337.83 | 59.156 | 0.3 | 83.746 | 73.819 | 64.885 | 51.181 | 130.38 | 106.48 | 89.105 | 61.719 | 276.26 | 184.20 | 143.95 | 73.113 | 3013.4 | 444.94 | 517.82 | 99.809 | 0.4 | 83.746 | 71.006 | 67.390 | 49.118 | 130.38 | 100.34 | 94.387 | 57.872 | 276.26 | 165.85 | 159.57 | 65.129 | 3013.4 | 333.07 | 741.94 | 184.59 | 0.5 | 83.746 | 68.397 | 69.958 | 47.119 | 130.38 | 94.868 | 99.874 | 54.230 | 276.26 | 150.84 | 176.29 | 58.236 | 3013.4 | 261.12 | 1010.2 | 313.50 | 0.6 | 83.746 | 65.970 | 72.589 | 45.182 | 130.38 | 89.957 | 105.57 | 50.793 | 276.26 | 138.30 | 194.10 | 52.434 | 3013.4 | 211.39 | 1322.6 | 486.54 | 0.7 | 83.746 | 63.707 | 75.284 | 43.309 | 130.38 | 85.526 | 111.46 | 47.561 | 276.26 | 127.67 | 213.00 | 47.725 | 3013.4 | 175.28 | 1679.1 | 703.71 | 0.8 | 83.746 | 61.592 | 78.041 | 41.498 | 130.38 | 81.506 | 117.56 | 44.534 | 276.26 | 118.54 | 233.00 | 44.107 | 3013.4 | 148.06 | 2079.7 | 965.01 | 0.9 | 83.746 | 59.610 | 80.862 | 39.751 | 130.38 | 77.842 | 123.87 | 41.712 | 276.26 | 110.59 | 254.08 | 41.581 | 3013.4 | 126.97 | 2524.5 | 1270.4 | 1.0 | 83.746 | 57.749 | 83.746 | 38.067 | 130.38 | 74.490 | 130.38 | 39.095 | 276.26 | 103.62 | 276.26 | 40.147 | 3013.4 | 110.24 | 3013.4 | 1620.0 |
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