1

Fig. 1. A sample graph of the double normal distribution with m = 22, s1 = 4 and s2 = 7. Notation: m – the mean of the compound distributions that becomes an overall distribution mode, s1 – the standard deviation of the first compound distribution, which makes a left half of the overall distribution, s2 – the standard deviation of the second compound distribution, which makes a right half of the overall distribution; solid lines show the resulting double normal distribution.

Table 1. Summary statistics of the 20 sample plots used in the study.
  Area Age N QMD MD HL BA SI
[ha] [years] [trees] [cm] [cm] [m] [m2 ha–1] [m]
Minimum 0.049 26 258 6.2 5.9 7.70 22.40 16.90
Maximum 0.810 123 500 28.7 28.3 23.20 31.20 27.20
Average 0.297 58 378 15.5 15.0 14.34 27.18 22.03
Std dev. 0.235 25 67 6.0 5.9 3.70 2.55 3.53
Notation: Area – size of the sample plot; Age – average stand age; N – number of trees per plot; QMD – quadratic mean diameter of the stand (calculated from BA and N); MD – mean diameter (arithmetic mean of breast height diameter); HL – average stand height calculated using the Lorey formula (basal area weighted average height); BA – basal area per hectare; SI – base-age 100 site index (Bruchwald et al. 2000; Cieszewski and Zasada 2002).
2

Fig. 2. A sample graph of the double normal distribution (solid line) outperforming the Weibull distribution (dashed line) for the plot BT119 (histogram).

3

Fig. 3. A sample graph of the Weibull distribution (dashed line) outperforming the double normal distribution (solid line) for the plot BT130 (histogram).

Table 2. Detailed values of the Dn statistics for various distributions and methods of fitting. MLE denotes the maximum likelihood method, MOM3 – method of moments based on first three moments of the sample, MOM2 – method of moments based on first two moments of the sample and the distribution mode estimated using the approximate formula: mode = mean – 4.8 · (mean – median), MOM2W – method of moments based on first two moments of the sample. Lower value of the Dn statistics denote better fit. The best fit is marked with bold numbers.
    Double normal Weibull
Plot MLE MOM3 a) MOM2 a) MLE MOM2W
BT112 0.0356 0.0327 0.0327 0.0324 0.0317
BT113 0.0347 0.0285 0.0285 0.0294 0.0282
BT114 0.0452 0.0542 0.0542 0.0568 0.0613
BT115 0.0391 0.0490 0.0490 0.0373 0.0412
BT116 0.0385 0.0413 0.0413 0.0422 0.0428
BT117 0.0344 0.0355 0.0355 0.0380 0.0378
BT118 0.0453 0.0294 0.0294 0.0255 0.0249
BT119 0.0244 0.0399 0.0399 0.0384 0.0343
BT120 0.0572 0.0396 0.0396 0.0345 0.0321
BT121 0.0277 0.0441 0.0441 0.0379 0.0375
BT122 0.0317 0.0355 0.0355 0.0310 0.0310
BT123 0.0315 0.0310 0.0310 0.0212 0.0211
BT124 0.0253 0.0197 0.0197 0.0275 0.0271
BT125 0.0221 0.0261 0.0261 0.0474 0.0406
BT126 0.0394 0.0476 0.0476 0.0552 0.0462
BT127 0.0322 0.0283 0.0283 0.0264 0.0302
BT128 0.0340 0.0323 0.0323 0.0289 0.0298
BT129 0.0391 0.0349 0.0349 0.0397 0.0386
BT130 0.0361 0.0354 0.0354 0.0273 0.0277
BT131 0.0499 0.0390 0.0390 0.0315 0.0270
Avg. 0.0362 0.0362 0.0362 0.0354 0.0346
a) Values of the Dn statistics for MOM2 and MOM3 are not identical, but the differences exist below the resolution of the presentation of results.
Table 3. Detailed values of the Dn statistics and number of cases when the hypothesis about matching the empirical and theoretical distributions was rejected for various distributions and methods of fitting for 100 samples of 50 trees. For explanations of the abbreviations see Table 2. MLE(S) – the maximum likelihood method with simulated annealing optimization algorithm. The best fit without MLE(S) method is marked with bold numbers, and with the MLE(S) – with underlined numbers.
Plot Double normal Weibull
MLE 0.05 MLE(S) 0.05 MOM3 0.05 MOM2 0.05 MLE 0.05 MOM2W 0.05
BT112 0.0973 7 0.0753 0 0.0811 0 0.0890 0 0.0674 0 0.0660 0
BT113 0.0967 8 0.0715 0 0.0698 0 0.0756 0 0.0670 0 0.0652 0
BT114 0.1054 7 0.0845 4 0.1533 0 0.0859 0 0.0855 1 0.0894 0
BT115 0.0906 2 0.0798 2 0.0985 0 0.0793 0 0.0802 0 0.0771 0
BT116 0.1329 12 0.0752 0 0.0733 0 0.0789 0 0.0670 0 0.0729 0
BT117 0.1352 14 0.0773 0 0.0867 0 0.0847 0 0.0757 0 0.0752 0
BT118 0.1168 9 0.0773 1 0.0691 0 0.0741 0 0.0697 0 0.0711 0
BT119 0.1078 6 0.0776 0 0.1251 0 0.0910 0 0.0789 0 0.0781 0
BT120 0.1563 18 0.0799 0 0.0877 0 0.0900 0 0.0758 0 0.0794 0
BT121 0.1193 12 0.0720 0 0.0753 0 0.0791 0 0.0687 1 0.0706 0
BT122 0.1738 20 0.0735 0 0.0785 0 0.0776 0 0.0660 0 0.0690 0
BT123 0.1794 22 0.0726 0 0.0890 0 0.0753 0 0.0627 0 0.0608 0
BT124 0.1350 15 0.0718 1 0.0978 0 0.0693 0 0.0703 2 0.0683 0
BT125 0.0756 2 0.0857 1 0.0743 0 0.0726 0 0.0768 1 0.0747 0
BT126 0.0825 1 0.0750 1 0.0896 0 0.0819 0 0.0889 2 0.0914 0
BT127 0.0838 2 0.0709 0 0.0783 0 0.0829 0 0.0723 0 0.0700 0
BT128 0.1013 5 0.0740 0 0.0778 0 0.0830 0 0.0684 0 0.0748 0
BT129 0.0971 5 0.0735 0 0.0855 0 0.0886 0 0.0769 1 0.0683 0
BT130 0.0946 5 0.0767 0 0.0697 0 0.0774 0 0.0716 0 0.0743 0
BT131 0.1133 8 0.0780 0 0.0912 0 0.0829 0 0.0727 0 0.0769 0
Avg. 0.1147   0.0761   0.0876   0.0809   0.0731   0.0737  
Table 4. Detailed values of the Dn statistics and number of cases when the hypothesis about matching the empirical and theoretical distributions was rejected for various distributions and methods of fitting for 100 samples of 10 trees. For explanations of the abbreviations see Table 2. The best fit is marked with bold numbers.
Plot Double normal Weibull
MLE 0.05 MLE(S) 0.05 MOM3 0.05 MOM2 0.05 MLE 0.05 MOM2W 0.05
BT112 0.3519 45 0.1857 6 0.1678 1 0.1755 1 0.1472 0 0.1391 0
BT113 0.3210 37 0.2050 12 0.1555 2 0.1562 1 0.1387 0 0.1379 1
BT114 0.4111 53 0.2461 15 0.2076 4 0.1735 0 0.1577 0 0.1590 1
BT115 0.3605 39 0.1953 4 0.1823 2 0.1675 0 0.1482 0 0.1529 0
BT116 0.3478 40 0.1891 3 0.1745 2 0.1698 1 0.1394 0 0.1408 0
BT117 0.3947 50 0.2154 9 0.1894 4 0.1786 3 0.1528 2 0.1466 0
BT118 0.3348 38 0.1868 3 0.1530 2 0.1556 1 0.1478 0 0.1407 2
BT119 0.3604 44 0.2275 7 0.1867 3 0.1722 3 0.1536 0 0.1499 1
BT120 0.4047 50 0.2154 7 0.1698 3 0.1708 2 0.1510 1 0.1466 0
BT121 0.3522 42 0.2059 3 0.1690 1 0.1724 1 0.1290 0 0.1358 0
BT122 0.4264 56 0.1979 9 0.1782 2 0.1688 0 0.1473 0 0.1362 1
BT123 0.3514 39 0.2031 9 0.1742 1 0.1567 1 0.1419 0 0.1382 1
BT124 0.3946 50 0.1671 2 0.1657 1 0.1576 0 0.1325 0 0.1350 0
BT125 0.3530 40 0.1739 4 0.1642 2 0.1677 2 0.1611 0 0.1485 1
BT126 0.3227 36 0.1791 3 0.1806 3 0.1619 0 0.1636 0 0.1536 0
BT127 0.3840 49 0.1700 0 0.1451 0 0.1478 0 0.1565 0 0.1488 0
BT128 0.3413 38 0.1839 1 0.1616 5 0.1607 3 0.1626 1 0.1402 0
BT129 0.3259 35 0.1923 6 0.1644 2 0.1725 1 0.1648 0 0.1388 0
BT130 0.3322 38 0.1894 7 0.1716 2 0.1812 1 0.1668 1 0.1447 2
BT131 0.3324 36 0.1849 4 0.1669 1 0.1636 1 0.1586 2 0.1524 2
Avg. 0.3602   0.1958   0.1714   0.1665   0.1511   0.1443