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

Table 8.

Estimated MSE for different estimators at p = 9 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 7.06866 6.52061 6.01562 6.44942 4.25378 3.5949 3.02618 1.42379 0.97861 3.66112 1.52125
75 2.04102 1.77469 1.56616 1.78387 1.1806 0.98454 0.78045 0.24191 0.15483 0.94489 0.26642
150 0.79185 0.74977 0.71275 0.74773 0.51018 0.42981 0.34364 0.10325 0.06801 0.39445 0.11383
200 0.47735 0.46198 0.44757 0.45771 0.3103 0.26314 0.21152 0.06693 0.04846 0.23778 0.07294
300 0.36382 0.3563 0.34909 0.35324 0.24516 0.20947 0.16983 0.0543 0.03974 0.18639 0.05916
400 0.23405 0.22976 0.2256 0.22736 0.1501 0.12686 0.10041 0.03031 0.0261 0.10826 0.03244
0.85 30 12.2065 11.96343 9.22308 10.20689 5.29031 4.58022 3.59615 1.396 0.82003 3.43257 1.27171
75 2.15174 1.93216 1.76066 1.97197 1.36983 1.15279 0.93559 0.32023 0.20626 1.11523 0.35115
150 0.99095 0.93454 0.88553 0.93354 0.64419 0.54648 0.44385 0.15374 0.10228 0.51256 0.16767
200 0.66282 0.63483 0.60928 0.63056 0.42024 0.35439 0.28219 0.0811 0.05387 0.32245 0.08974
300 0.46491 0.45237 0.4405 0.44864 0.30967 0.26296 0.21237 0.06424 0.04226 0.23762 0.07101
400 0.3686 0.35947 0.35071 0.35571 0.23483 0.19676 0.15513 0.04161 0.02914 0.17084 0.04604
0.90 30 16.1216 14.12896 12.38458 13.55324 6.62773 5.25457 4.16014 1.57106 0.9913 5.48195 1.70696
75 3.15944 2.71129 2.34182 2.74773 1.69349 1.3901 1.07534 0.30093 0.18758 1.34573 0.33379
150 1.36055 1.26435 1.18044 1.27819 0.87593 0.73646 0.59065 0.18277 0.11283 0.69982 0.20213
200 1.22283 1.10382 1.00116 1.1175 0.70701 0.57373 0.43502 0.09034 0.05384 0.52193 0.10317
300 0.70297 0.66275 0.62604 0.65764 0.39805 0.32487 0.24441 0.05014 0.03332 0.28604 0.05662
400 0.49001 0.47002 0.45166 0.46402 0.29606 0.24658 0.19181 0.04826 0.03177 0.2169 0.05384
0.95 30 57.5364 50.65413 45.10653 47.68283 16.88082 12.80335 9.99353 3.85353 2.51441 14.40752 4.16851
75 6.07069 5.34264 4.71659 5.37574 3.32054 2.72268 2.14339 0.71872 0.46314 2.75744 0.78464
150 3.11469 2.79998 2.5222 2.86519 1.76555 1.42049 1.08569 0.26084 0.14953 1.39067 0.295
200 2.2451 1.94156 1.69362 1.98443 1.17039 0.94121 0.70439 0.15085 0.08831 0.89358 0.17148
300 1.14368 1.03288 0.94162 1.03812 0.63089 0.51417 0.38779 0.08171 0.04835 0.4712 0.09305
400 0.95289 0.90168 0.85529 0.90192 0.598 0.49409 0.38736 0.1001 0.05768 0.45761 0.11276
0.99 30 108.40914 101.72837 95.56783 100.73019 56.64332 45.47928 36.82599 15.56412 10.22591 51.99598 16.75204
75 34.77464 30.75336 27.24856 30.51649 16.1249 12.52277 9.30831 2.29935 1.28055 13.95642 2.58621
150 19.55027 16.98758 14.74125 16.98293 9.065 7.00506 5.14027 1.14291 0.59838 7.65428 1.30046
200 10.72963 8.95639 7.4008 9.18044 4.64182 3.49441 2.42921 0.36963 0.20193 3.6507 0.43094
300 6.71319 5.63807 4.69959 5.78008 3.15813 2.44236 1.75644 0.3161 0.16521 2.47979 0.36583
400 4.92142 4.18446 3.53909 4.36078 2.45692 1.88906 1.3606 0.24214 0.12627 1.88982 0.28059

Bolded values indicate the lowest MSE.