# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## Appendix B The Formula for Sample L-Moments  
For calculating the sample L-moments, the observed data \(x_{i}\) must be in ascending order. The sample L-moments are:  
\[ \begin{array}{ll}L_{1}=b_{0}&-\text{ first L -- moment}\\ L_{2}=2\cdot b_{1}-b_{0}&-\text{ second L -- moment}\\ L_{3}=6\cdot b_{2}-6\cdot b_{1}+b_{0}&-\text{ third L -- moment}\\ L_{4}=20\cdot b_{3}-30\cdot b_{2}+12\cdot b_{1}-b_{0}&-\text{ fourth L -- moment}\end{array} \]  
where \(b_{0},b_{1},b_{2},b_{3}\) represent natural estimators, expressed as:  
\[b_{0}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\]\[b_{1}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\]  
\[b_{2}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\cdot\frac{i-2}{n-2}\]  
\[b_{3}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\cdot\frac{i-2}{n- 2}\cdot\frac{i-3}{n-3}\]  
where \(n\) is the length of data series. Using the L-moments, the coefficients of variation, (\(\tau_{2}\)), skewness (\(\tau_{3}\)) and kurtosis (\(\tau_{4}\)) are defined as [14; 15; 16; 17]:  
\(\tau_{2}=\frac{L_{2}}{L_{1}}\)--is the L-CV;  
\(\tau_{3}=\frac{L_{3}}{L_{2}}\)--is the L-Skewness;  
\(\tau_{4}=\frac{L_{4}}{L_{2}}\)--is the L-Kurtosis  
In the next section, we present the variation of L-kurtosis, depending on the positive L-skewness, obtained with the L-moments method, for certain theoretical distributions often used in hydrology and in this article. Pearson III:  
\(\tau_{4}=0.121775+0.030285\cdot\tau_{3}+0.0266125\cdot\tau_{3}^{2}+0.8774691 \cdot\tau_{3}^{3}-0.0564795\cdot\tau_{3}^{4}\)  
Pearson V:  
\(\tau_{4}=0.1089545-0.1542626\cdot\tau_{3}+1.0657605\cdot\tau_{3}^{2}-0.3521005 \cdot\tau_{3}^{3}+0.3269967\cdot\tau_{3}^{4}\)  
Wilson-Hilferty:  
\(\tau_{4}=0.1177849-0.5367173\cdot\tau_{3}+1.4180786\cdot\tau_{3}^{2}-0.208469 7\cdot\tau_{3}^{3}+0.2098975\cdot\tau_{3}^{4}\)  
CHI:  
\(\tau_{4}=0.1274475-0.2174617\cdot\tau_{3}-0.0945508\cdot\tau_{3}^{2}+2.657290 5\cdot\tau_{3}^{3}-2.369862\cdot\tau_{3}^{4}+0.9016064\cdot\tau_{3}^{5}\)  
Inverse CHI:  
\(\tau_{4}=0.1215494+0.0260015\cdot\tau_{3}+0.6839989\cdot\tau_{3}^{2}+2.34321 88\cdot\tau_{3}^{3}-7.9178585\cdot\tau_{3}^{4}+11.9165941\cdot\tau_{3}^{5}-\)  
\(8.06007\cdot\tau_{3}^{6}+1.8820702\cdot\tau_{3}^{7}\)  
Pseudo-Weibull:  
\(\tau_{4}=0.1132189-0.1242052\cdot\tau_{3}+1.1329458\cdot\tau_{3}^{2}-0.4716246 \cdot\tau_{3}^{3}+0.3449906\cdot\tau_{3}^{4}\)Wakeby:  
\[\tau_{4}=-0.07347+0.14443\cdot\tau_{3}+1.03879\cdot\tau_{3}^{2}-0.14602\cdot\tau_{ 3}^{3}+0.03357\cdot\tau_{3}^{4}\]  
Pareto:  
\[\tau_{4}=-0.0003668+0.2070484\cdot\tau_{3}+0.9264\cdot\tau_{3}^{2}-0.133564\cdot \tau_{3}^{3}\]  
Generalized Extreme Value:  
\[\tau_{4}=0.1072214+0.1143838\cdot\tau_{3}+0.8341466\cdot\tau_{3}^{2}-0.0632425 \cdot\tau_{3}^{3}+0.0074607\cdot\tau_{3}^{4}\]  
Frechet:  
\[\tau_{4}=0.1069938+0.1155235\cdot\tau_{3}+0.8294258\cdot\tau_{3}^{2}-0.0528083 \cdot\tau_{3}^{3}\]  
Weibull:  
\[\tau_{4}=0.1057425-0.0753465\cdot\tau_{3}+0.6176919\cdot\tau_{3}^{2}+0.506512 7\cdot\tau_{3}^{3}-0.1788008\cdot\tau_{3}^{4}\]  
Log-Normal:  
\[\tau_{4}=0.1238145-0.032954\cdot\tau_{3}+0.9783895\cdot\tau_{3}^{2}-0.3929245 \cdot\tau_{3}^{3}+0.3174611\cdot\tau_{3}^{4}\]  
Log-Logistic:  
\[b_{1}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\]0  
Paralogistic:  
\[b_{1}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\]1  
InverseParalogistic:  
\[b_{1}=\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}\cdot\frac{i-1}{n-1}\]2