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Table 1 GBL fit for first digit of powerful integer powers: MAD vs. WLS criterion

From: A first digit theorem for powerful integer powers

s = 1 Parameters Δ to LL estimate MAD GoF measures WLS GoF measures
m = WLS MAD WLS MAD LL WLS MAD LL WLS MAD
8 0.50630179 0.50602471 1.819 1.947 282.9 135.2 134.6 90.14 28.44 28.75
9 0.50398677 0.50439385 2.298 1.890 168.3 90.24 86.04 66.89 21.19 22.62
10 0.50258542 0.50277813 2.755 2.340 92.13 45.33 45.02 42.88 12.36 13.05
11 0.50170493 0.50180536 3.116 2.650 44.37 17.57 16.34 22.63 4.505 4.911
12 0.50116789 0.50121344 2.932 2.477 19.77 8.163 7.927 8.998 1.546 1.726
13 0.50080206 0.50083278 2.077 1.415 6.993 4.335 4.077 2.519 0.784 0.960
14 0.50054625 0.50054722 0.289 0.244 1.821 1.802 1.794 0.345 0.330 0.330
15 0.50037082 0.50036094 3.110 2.122 2.042 1.252 1.200 1.154 0.315 0.400
s = 2 Parameters Δ to LL estimate MAD GoF measures WLS GoF measures
m = WLS MAD WLS MAD LL WLS MAD LL WLS MAD
8 0.25428436 0.25324296 0.383 0.867 238.4 218.8 193.9 163.9 157.8 167.6
9 0.25428436 0.25267279 1.142 0.469 76.66 122.8 59.30 28.98 66.02 28.52
10 0.25166049 0.25155591 0.585 0.811 26.64 19.31 18.58 7.514 4.409 4.870
11 0.25096820 0.25089373 1.021 1.366 22.15 15.50 13.56 10.95 6.568 7.071
12 0.25062939 0.25059921 1.012 1.314 11.19 8.423 7.972 6.168 4.169 4.347
13 0.25042363 0.25041307 0.552 0.779 3.923 3.055 2.805 1.563 1.287 1.334
14 0.25028419 0.25028723 0.369 0.510 1.533 1.302 1.238 0.540 0.482 0.491
15 0.25018970 0.25019163 1.984 2.177 1.344 0.640 0.624 1.027 0.258 0.266
s = 3 Parameters Δ to LL estimate MAD GoF measures WLS GoF measures
m = WLS MAD WLS MAD LL WLS MAD LL WLS MAD
8 0.16900766 0.16723059 0.495 1.319 274.3 257.7 248.0 575.4 553.1 615.3
9 0.16801302 0.16762481 0.748 1.137 116.4 89.18 81.67 142.8 119.1 125.5
10 0.16769485 0.16753300 0.560 0.909 60.96 58.26 57.25 124.0 117.8 120.2
11 0.16729754 0.16730322 0.748 0.722 18.31 13.37 13.25 24.45 19.34 19.35
12 0.16710133 0.16708812 0.524 0.656 7.100 5.652 5.371 6.058 4.895 4.969
13 0.16695408 0.16695069 0.260 0.333 3.798 3.374 3.255 3.664 3.530 3.541
14 0.16685701 0.16685618 0.287 0.249 2.484 2.358 2.338 3.931 3.856 3.857
15 0.16679314 0.16679442 1.323 1.452 1.321 0.972 0.952 2.021 1.279 1.286
s = 4 Parameters Δ to LL estimate MAD GoF measures WLS GoF measures
m = WLS MAD WLS MAD LL WLS MAD LL WLS MAD
8 0.12774699 0.12760514 0.089 0.023 336.0 339.0 334.9 1704 1702 1703
9 0.12682400 0.12682674 0.253 0.256 108.7 100.3 100.2 495.4 489.5 489.6
10 0.12592594 0.12589200 0.086 0.160 38.75 37.58 36.59 149.2 148.9 149.1
11 0.12543815 0.12539409 0.724 0.928 21.98 16.83 15.38 52.95 42.62 43.45
12 0.12530767 0.12526505 0.576 1.002 7.884 6.275 5.841 18.01 14.97 16.63
13 0.12521961 0.12520948 0.108 0.326 4.885 4.688 4.291 20.84 20.79 20.99
14 0.12514734 0.12514391 0.428 0.269 1.315 1.279 1.181 4.693 4.331 4.381
15 0.12509739 0.12509771 1.246 1.278 0.940 0.587 0.579 3.039 1.619 1.620
s = 5 Parameters Δ to LL estimate MAD GoF measures WLS GoF measures
m = WLS MAD WLS MAD LL WLS MAD LL WLS MAD
8 0.10483305 0.10494392 1.295 1.346 551.6 498.4 496.2 8531 7818 7819
9 0.10110871 0.10166842 0.148 0.412 182.7 185.4 175.0 2055 2051 2113
10 0.10041545 0.10050001 0.770 0.588 39.74 30.51 28.02 211.5 157.1 160.2
11 0.10035943 0.10038750 0.538 0.407 21.67 19.45 18.85 124.1 111.8 112.5
12 0.10021129 0.10018888 0.809 1.033 10.05 8.099 7.558 60.76 47.82 48.81
13 0.10015612 0.10013482 0.508 0.967 5.373 4.752 4.238 35.74 33.38 35.31
14 0.10011058 0.10010322 0.004 0.337 1.790 1.792 1.653 9.827 9.827 10.32
15 0.10007403 0.10007147 0.609 0.353 0.794 0.713 0.655 4.477 3.745 3.874
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