Research Article
More on the Ridge Parameter Estimators for the Gamma Ridge Regression Model: Simulation and Applications
| | Estimators | | | | | | n = 25 | n = 50 | n = 100 | n = 200 | n = 25 | n = 50 | n = 100 | n = 200 | n = 25 | n = 50 | n = 100 | n = 200 | n = 25 | n = 50 | n = 100 | n = 200 |
| | MLE | 9.6010 | 5.5618 | 3.2084 | 2.0371 | 3561.1 | 1292.6 | 593.50 | 300.20 | 5593.8 | 2130.6 | 955.78 | 499.15 | 3.33 × 106 | 346970 | 107872 | 53714 | | k1 | 3.1043 | 2.1001 | 1.5174 | 1.2094 | 782.84 | 322.77 | 159.97 | 83.939 | 1202.1 | 513.88 | 244.16 | 129.66 | 571703 | 76371 | 25083 | 14025 | | k2 | 4.3616 | 4.3566 | 3.0406 | 2.0163 | 4.9601 | 10.129 | 19.952 | 40.375 | 4.9244 | 9.3410 | 16.7191 | 33.1176 | 2.8574 | 12.103 | 34.454 | 64.003 | | k3 | 1.6842 | 1.7262 | 1.6810 | 1.5072 | 2.0512 | 3.2449 | 4.6698 | 5.9456 | 2.0958 | 3.2448 | 4.5307 | 5.5417 | 0.9532 | 1.2771 | 2.5256 | 6.0806 | | k4 | 1.2958 | 1.1306 | 1.0311 | 0.9515 | 40.032 | 27.581 | 23.541 | 18.529 | 32.743 | 23.461 | 20.900 | 17.567 | 837.82 | 312.66 | 225.79 | 186.29 | | k5 | 1.1874 | 1.2130 | 1.2455 | 1.2306 | 20.306 | 21.391 | 24.858 | 27.227 | 16.706 | 16.312 | 18.478 | 19.750 | 289.07 | 188.86 | 192.34 | 191.94 | | k6 | 8.1331 | 5.2637 | 3.1424 | 2.0243 | 2566.8 | 1176.5 | 575.40 | 296.84 | 4112.7 | 1946.3 | 926.82 | 493.41 | 1.64 × 106 | 293679 | 102999 | 52983 | | k7 | 6.8435 | 4.7878 | 3.0151 | 1.9889 | 1737.9 | 965.71 | 520.59 | 282.35 | 2547.1 | 1502.7 | 809.86 | 460.62 | 682778 | 174043 | 77650 | 46440 | | k8 | 0.9059 | 0.9457 | 1.0191 | 1.1058 | 0.9040 | 1.0541 | 1.9776 | 3.8167 | 0.9023 | 0.9689 | 1.6650 | 3.1905 | 0.9898 | 0.9638 | 0.9130 | 0.9046 | | k9 | 6.8435 | 4.7878 | 3.0151 | 1.9889 | 1737.9 | 965.71 | 520.59 | 282.35 | 2547.1 | 1502.7 | 809.86 | 460.62 | 682778 | 174043 | 77650 | 46440 | | k10 | 1.2934 | 1.2445 | 1.1761 | 1.1038 | 10.714 | 11.624 | 12.091 | 11.380 | 11.475 | 11.584 | 11.741 | 10.680 | 88.500 | 97.224 | 106.04 | 103.09 | | k11 | 1.0164 | 1.1133 | 1.2295 | 1.2746 | 1.0753 | 1.7727 | 3.4280 | 5.2979 | 1.0191 | 1.5517 | 2.9719 | 4.6412 | 0.9703 | 0.9367 | 1.0157 | 1.8157 | | k12 | 1.0164 | 1.1133 | 1.2295 | 1.2746 | 1.0753 | 1.7727 | 3.4280 | 5.2979 | 1.0191 | 1.5517 | 2.9719 | 4.6412 | 0.9703 | 0.9367 | 1.0157 | 1.8157 | | k13 | 1.0540 | 1.1044 | 1.1400 | 1.1433 | 4.0045 | 5.5994 | 7.2317 | 8.2766 | 3.9546 | 5.3270 | 6.7355 | 7.4109 | 7.2765 | 16.093 | 29.684 | 44.530 | | k14 | 1.0234 | 1.1004 | 1.1764 | 1.2087 | 1.4841 | 2.6663 | 4.3538 | 5.8500 | 1.4236 | 2.5088 | 4.0672 | 5.3404 | 0.9066 | 1.0717 | 2.3798 | 6.9137 | | k15 | 1.1875 | 1.2328 | 1.2422 | 1.2096 | 5.1530 | 6.7637 | 8.2726 | 9.0279 | 5.3042 | 6.6085 | 7.8559 | 8.1484 | 15.495 | 26.677 | 41.192 | 54.775 | | k16 | 1.0299 | 1.1092 | 1.1847 | 1.2094 | 1.5038 | 2.7141 | 4.4252 | 5.9113 | 1.4331 | 2.5337 | 4.1159 | 5.3926 | 0.9108 | 1.0669 | 2.4040 | 7.0402 | | k17 | 8.9739 | 5.4007 | 3.1728 | 2.0295 | 1220.9 | 668.02 | 404.22 | 242.28 | 1937.9 | 1125.9 | 655.00 | 405.83 | 244547 | 52120 | 26410 | 20091 | | k18 | 2.7186 | 2.5233 | 2.1580 | 1.7154 | 104.20 | 106.64 | 120.01 | 113.61 | 73.169 | 74.856 | 93.473 | 104.84 | 1358.2 | 567.71 | 634.90 | 1010.5 | | k19 | 6.8435 | 4.7878 | 3.0151 | 1.9889 | 1737.9 | 965.71 | 520.59 | 282.35 | 2547.1 | 1502.7 | 809.86 | 460.62 | 682778 | 174043 | 77650 | 46440 | | k20 | 0.9842 | 0.9715 | 0.9651 | 0.9441 | 4.4560 | 5.5242 | 6.6770 | 7.1697 | 4.5022 | 5.4169 | 6.3723 | 6.7603 | 17.204 | 24.371 | 35.986 | 47.610 | | k21 | 1.8673 | 1.4341 | 1.1900 | 1.0411 | 221.81 | 108.15 | 65.324 | 39.245 | 274.96 | 141.01 | 85.140 | 50.996 | 70721 | 12029 | 4522.0 | 3056.1 | | k22 | 1.3442 | 1.2811 | 1.2129 | 1.1535 | 12.475 | 13.124 | 13.059 | 12.068 | 13.657 | 13.269 | 13.062 | 11.539 | 120.54 | 127.05 | 126.51 | 118.24 | | k23 | 0.9082 | 0.8692 | 0.8360 | 0.8292 | 1.0043 | 1.0037 | 1.0023 | 0.9994 | 1.0046 | 1.0042 | 1.0034 | 1.0016 | 1.0050 | 1.0050 | 1.0050 | 1.0050 | | k24 | 1.1874 | 1.2130 | 1.2455 | 1.2306 | 20.306 | 21.391 | 24.858 | 27.227 | 16.706 | 16.312 | 18.478 | 19.750 | 289.07 | 188.86 | 192.34 | 191.94 | | k25 | 0.8493 | 0.8327 | 0.8333 | 0.8440 | 0.9979 | 0.9928 | 0.9823 | 0.9647 | 0.9998 | 0.9958 | 0.9876 | 0.9732 | 1.0049 | 1.0048 | 1.0046 | 1.0044 | | k26 | 6.8435 | 4.7878 | 3.0151 | 1.9889 | 1737.9 | 965.71 | 520.59 | 282.35 | 2547.1 | 1502.7 | 809.86 | 460.62 | 682778 | 174043 | 77650 | 46440 | | k27 | 0.8342 | 0.8154 | 0.8059 | 0.8028 | 0.9750 | 0.9743 | 0.9723 | 0.9698 | 0.9761 | 0.9756 | 0.9744 | 0.9729 | 0.9775 | 0.9775 | 0.9775 | 0.9774 | | k28 | 3.4675 | 2.6273 | 1.8240 | 1.3735 | 4.1485 | 8.3990 | 14.407 | 22.436 | 4.2017 | 8.0180 | 12.895 | 19.969 | 1.4077 | 3.5905 | 13.339 | 34.803 | | k29 | 1.0432 | 1.0860 | 1.1394 | 1.1339 | 2.0017 | 3.1462 | 4.7737 | 6.0718 | 1.9879 | 3.0990 | 4.5283 | 5.6102 | 1.6913 | 3.6085 | 9.0947 | 19.250 | | k30 | 1.2608 | 1.3681 | 1.4243 | 1.3742 | 2.9601 | 4.5930 | 6.4490 | 7.7522 | 3.0592 | 4.5416 | 6.1564 | 7.0738 | 3.3642 | 7.6789 | 17.446 | 32.518 | | k31 | 1.2192 | 1.2911 | 1.3315 | 1.2936 | 2.8558 | 4.2696 | 6.0127 | 7.3217 | 2.8536 | 4.1842 | 5.7144 | 6.6185 | 2.8315 | 5.8035 | 13.853 | 26.954 | | k32 | 0.9184 | 0.9260 | 0.9609 | 0.9726 | 1.3047 | 1.9550 | 3.1538 | 4.3716 | 1.2783 | 1.9420 | 3.0184 | 4.1632 | 1.0588 | 1.6685 | 3.9515 | 9.3715 | | k33 | 1.0710 | 1.1241 | 1.1561 | 1.1498 | 3.0687 | 4.6443 | 6.0580 | 7.1013 | 2.9940 | 4.4092 | 5.6992 | 6.4764 | 2.8571 | 7.2544 | 16.207 | 28.323 | | k34 | 0.8647 | 0.8509 | 0.8468 | 0.8524 | 1.0809 | 1.5181 | 2.2975 | 3.2275 | 1.0567 | 1.4496 | 2.2259 | 3.0796 | 0.9280 | 0.9858 | 1.5571 | 3.1199 | | k35 | 0.9145 | 0.9120 | 0.9063 | 0.9035 | 1.8677 | 2.9284 | 4.2570 | 5.3793 | 1.7451 | 2.6994 | 3.9281 | 4.8864 | 1.1317 | 1.9477 | 4.9791 | 10.620 | | k36 | 2.5126 | 2.6238 | 2.3339 | 1.8341 | 3.9583 | 5.8959 | 7.7821 | 9.1872 | 4.2735 | 5.9296 | 7.5911 | 8.4059 | 4.8182 | 9.8125 | 19.279 | 34.599 | | k37 | 1.5227 | 1.5675 | 1.4751 | 1.3456 | 6.2225 | 9.5791 | 10.522 | 10.626 | 5.9614 | 7.5162 | 9.3481 | 10.500 | 7.2252 | 16.196 | 28.801 | 49.809 | | k38 | 1.0096 | 1.0181 | 1.0377 | 1.0524 | 3.1125 | 4.0882 | 5.2733 | 6.3137 | 3.0618 | 3.9437 | 4.9955 | 5.7973 | 5.0554 | 7.7808 | 13.520 | 23.133 | | k39 | 1.8673 | 1.4341 | 1.1900 | 1.0411 | 221.81 | 108.15 | 65.324 | 39.245 | 274.96 | 141.01 | 85.140 | 50.996 | 70721 | 12029 | 4522.0 | 3056.1 | | k40 | 1.0331 | 1.1508 | 1.2898 | 1.3433 | 1.0889 | 1.8249 | 3.4985 | 5.4011 | 1.0211 | 1.5951 | 3.0673 | 4.7291 | 0.9619 | 0.9160 | 0.9703 | 1.8369 |
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