| GI |
Gini index |
\[GI = \frac{2}{n}\frac{\sum_{i = 1}^{n}{i \cdot f_{i}}}{\sum_{i = 1}^{n}f_{i}} - \frac{n + 1}{n}\] |
[0, 1] |
| HH |
Herfindahl index |
\[HH = \frac{1}{n}\left\lbrack \frac{n\sum_{i = 1}^{n}\left( f_{i} - \overline{f} \right)^{2}}{\left( \sum_{i = 1}^{n}f_{i} \right)^{2}} + 1 \right\rbrack\] |
[1/n, 1] |
| HHm |
Herfindahl index (m) |
\[{HH}_{m} = \frac{1}{m}\left\lbrack \frac{n\sum_{i = 1}^{n}\left( f_{i} - \overline{f} \right)^{2}}{\left( \sum_{i = 1}^{n}f_{i} \right)^{2}} + 1 \right\rbrack\] |
[1/m, 1] |
| OM_PI |
Shannon entropy |
\[OM\_ PI = \sum_{i = 1}^{n}\left\lbrack \frac{f_{i}}{\sum_{j = 1}^{n}f_{j}}\log_{n}\left( \frac{\sum_{j = 1}^{n}f_{j}}{f_{i}} \right) \right\rbrack\] |
[0, 1] |
| OM_MI |
Shannon entropy (M) |
\[OM\_ MI = \sum_{i = 1}^{N}\left\{ \frac{f_{i}}{nM}\left\lbrack 1 + \ln\left( \frac{M}{f_{i}} \right) \right\rbrack \right\}\] |
[0, 1] |
| TH |
Theil index |
\[TH = \frac{1}{n}\sum_{i = 1}^{n}{\frac{f_{i}}{\overline{f}}\ln\left( \frac{f_{i}}{\overline{f}} \right)}\] |
[0, ln n] |
| DAL |
Dalton index |
\[DAL = 1 - \frac{\frac{1}{n}\sum_{i = 1}^{n}\left( {f_{i}}^{1 - \varepsilon} - 1 \right)}{{\overline{f}}^{1 - \varepsilon} - 1}\] |
[0, (1 − n−ε)/[1 − (f/n)ε−1]] |
| DALm |
Dalton index (m) |
\[{DAL}_{m} = 1 - \frac{\frac{1}{n}\sum_{i = 1}^{m}\left( f_{i}^{1 - \varepsilon} - 1 \right)}{\left( \frac{1}{m}\sum_{i = 1}^{m}f_{i} \right)^{1 - \varepsilon} - 1}\] |
[0, (n-1)/n] |
| ATK |
Atkinson index |
\[ATK = 1 - \left\lbrack \frac{1}{n}\sum_{i = 1}^{n}\left( \frac{f_{i}}{\overline{f}} \right)^{1 - \varepsilon} \right\rbrack^{\frac{1}{(1 - \varepsilon)}}\] |
[0, 1 − n−ε/(1−ε)] |