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. 2025 Aug 25;15:31162. doi: 10.1038/s41598-025-15334-9

Table 7.

Estimated MSE for different estimators at p = 6 and Inline graphic.

Inline graphic n PIGMLE PIGRRE PIGGRE
MLE Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic Inline graphic
0.80 30 7.18003 5.96725 4.95220 5.65691 3.43017 2.72024 2.06997 0.84094 0.53800 2.87566 0.96569
75 2.16899 1.49165 0.99617 1.36881 0.84479 0.68046 0.49076 0.18804 0.15851 0.61702 0.20495
150 0.86676 0.65707 0.52660 0.62939 0.39995 0.32912 0.24483 0.09814 0.08226 0.27872 0.10737
200 0.66804 0.61011 0.56204 0.60211 0.41545 0.34953 0.27908 0.12621 0.09206 0.31281 0.14130
300 0.40789 0.36347 0.33030 0.35021 0.22821 0.19257 0.15029 0.07255 0.06477 0.15840 0.07725
400 0.31116 0.29780 0.28569 0.29244 0.20175 0.17121 0.13784 0.06497 0.05267 0.14408 0.07151
0.85 30 10.14016 8.48593 7.04951 7.88894 4.06232 3.18657 2.45156 1.11959 0.77145 3.43226 1.25556
75 2.68506 1.90842 1.35977 1.77110 1.07970 0.86152 0.62508 0.22662 0.17170 0.80587 0.25533
150 0.98303 0.85630 0.75711 0.84855 0.55362 0.45568 0.35068 0.13520 0.09582 0.40985 0.15471
200 0.85518 0.73529 0.64939 0.72495 0.47859 0.39667 0.30818 0.12716 0.08925 0.35018 0.14409
300 0.60915 0.53869 0.48518 0.52807 0.34403 0.28468 0.22019 0.09294 0.07253 0.24267 0.10348
400 0.37681 0.35578 0.33698 0.34848 0.23258 0.19406 0.15270 0.06486 0.04946 0.16341 0.07279
0.90 30 13.14303 10.60679 8.52664 9.67515 5.15026 4.02647 3.06061 1.38919 0.97241 4.38099 1.55263
75 2.97547 2.46797 2.05120 2.50301 1.63325 1.32316 1.02410 0.42405 0.28170 1.31799 0.48440
150 1.64494 1.28766 1.03472 1.26495 0.79908 0.64378 0.47922 0.17359 0.12091 0.58764 0.19932
200 1.10051 0.95294 0.83500 0.95261 0.61667 0.50022 0.38336 0.14826 0.09859 0.45771 0.17084
300 0.82955 0.71539 0.62936 0.70889 0.45245 0.36832 0.28020 0.10575 0.07320 0.32251 0.12143
400 0.55415 0.51468 0.47972 0.50684 0.33374 0.27466 0.21368 0.08508 0.05906 0.23877 0.09744
0.95 30 47.89147 38.17942 30.49429 34.15357 14.82216 11.25518 8.38152 3.73163 2.37455 13.86783 4.20847
75 5.93897 4.43688 3.26921 4.14677 2.38095 1.85000 1.33513 0.45088 0.28393 1.89978 0.52735
150 3.77671 2.94719 2.29815 2.88179 1.62622 1.25264 0.89004 0.26519 0.16417 1.21939 0.31635
200 2.50137 1.79143 1.28696 1.73304 1.00482 0.78128 0.55048 0.16227 0.10631 0.72471 0.19204
300 1.39646 1.20607 1.05069 1.21629 0.77962 0.62152 0.46969 0.16892 0.10304 0.58134 0.19835
400 1.10468 0.91389 0.77171 0.91029 0.54009 0.42692 0.31007 0.09435 0.05997 0.37108 0.11227
0.99 30 104.99316 88.77856 75.12642 82.60415 33.77235 24.75321 18.05264 7.25678 4.23941 33.20595 8.37287
75 41.75106 32.20102 24.40079 28.96857 10.65210 7.39667 4.84938 1.27197 0.70639 9.76427 1.54692
150 15.15633 12.07267 9.49770 11.59891 5.29019 3.80320 2.55909 0.68119 0.37146 4.41116 0.83063
200 12.85437 10.69363 8.83174 10.47499 5.11777 3.71635 2.57294 0.76477 0.41932 4.22717 0.92073
300 8.18779 6.54940 5.18827 6.43319 3.18094 2.31601 1.58019 0.42582 0.22992 2.49836 0.52003
400 5.38889 4.35928 3.50626 4.33666 2.24028 1.63563 1.11685 0.28836 0.15509 1.69409 0.35542

Bolded values indicate the lowest MSE.