# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## Appendix B The Formula for Sample L-Moments
\begin{tabular}{c c} \hline
**Distribution** & **Skewness** \\ \hline Frechet & \(C_{s}=\frac{\Gamma\left(1-\frac{3}{2}\right)+2\cdot\Gamma\left(1-\frac{1}{2} \right)^{3}-3\cdot\Gamma\left(1-\frac{2}{2}+\right)\cdot\Gamma\left(1-\frac{1}{ 2}\right)}{\sqrt{\left(\Gamma\left(1-\frac{1}{2}\right)-\Gamma\left(1-\frac{1} {2}\right)^{2}\right)^{3}}}\) \\  & \(\alpha=\frac{1+0.441588375\cdot|\tau_{3}|\tau_{3}|-0.071007671\cdot\tau_{3}^{2}+ 0.015634368\cdot|\tau_{3}|^{3}}{-0.283759107+1.669931462\cdot|\tau_{3}|}\) \\ \hline Weibull & \(C_{s}=\frac{\Gamma\left(1+\frac{3}{2}\right)-3\cdot\Gamma\left(1+\frac{1}{2} \right)\cdot\Gamma\left(1+\frac{1}{2}\right)+2\cdot\Gamma\left(1+\frac{1}{2} \right)^{3}}{\left(\Gamma\left(1+\frac{3}{2}\right)-\Gamma\left(1+\frac{1}{2} \right)^{2}\right)^{3}}\) \\  & \(\alpha=\frac{3.528107902-6.294082546\cdot|\tau_{3}|+2767652838\cdot\tau_{3}^{2}} {1+4.59902049293\cdot|\tau_{3}|-79.93605172\cdot\tau_{3}^{2}+2.423742593\cdot| \tau_{3}|^{3}}\) \\ \hline Log-Logistic & \(C_{s}==\frac{\Gamma\left(1+\frac{3}{2}\right)\cdot\Gamma\left(1-\frac{3}{2} \right)+2\cdot\Gamma\left(1+\frac{3}{2}\right)\cdot\Gamma\left(1-\frac{1}{2} \right)^{3}\cdot\Gamma\left(1-\frac{3}{2}\right)\cdot\Gamma\left(1+\frac{3}{2} \right)}{\left[\Gamma\left(1+\frac{3}{2}\right)\cdot\Gamma\left(1-\frac{3}{2} \right)-\Gamma\left(1+\frac{1}{2}\right)^{2}\right]^{15}}}{\alpha\left(1+\frac {3}{2}\right)\cdot\Gamma\left(1-\frac{3}{2}\right)\cdot\Gamma\left(1-\frac{3 }{2}\right)^{2}\cdot\Gamma\left(1-\frac{3}{2}\right)^{2}\right]^{15}}\) \\  & \(\alpha=\frac{1}{7_{5}}\) \\ \hline Pareto & \(C_{s}=\frac{\alpha}{|\alpha|}\cdot\frac{2\cdot\left(-1\right)\cdot\left(2+1 \right)^{5}}{3+\alpha+1}\) \\  & \(\alpha=\frac{1-3\cdot\Gamma}{3\cdot\Gamma^{1}}\) \\ \hline Pearson V & \(\alpha=\exp\left(\begin{array}{c}0.692740135-0.466055586\cdot\ln(|\tau_{3}|)+0.460508385\cdot\ln(|\tau_{3}|)^{2}+\\ 0.470172127\cdot\ln(|\tau_{3}|)^{3}+0.938027763\cdot\ln(|\tau_{3}|)^{4}+\\ 0.805363294\cdot\ln(|\tau_{3}|)^{5}+0.359846034\cdot\ln(|\tau_{3}|)^{6}+\\ 0.08945247\cdot\ln(|\tau_{3}|)^{7}+0.011775528\cdot\ln(|\tau_{3}|)^{8}+0.00064166 4\cdot\ln(|\tau_{3}|)^{9}\end{array}\right)\) \\ \hline \(C_{s}=\frac{\alpha-\frac{3}{|\alpha|\alpha|}\cdot\Gamma\left(\alpha+\frac{3} {2}\right)\cdot\Gamma\left(\alpha+\frac{3}{2}\right)\cdot\Gamma\left(\alpha+ \frac{1}{2}\right)^{2}\cdot\Gamma\left(1+\frac{3}{2}\right)^{3}}{\sqrt{ \frac{\Gamma\left(\alpha+\frac{3}{2}\right)}{\Gamma\left(\alpha+\frac{3}{2} \right)}-\frac{\Gamma\left(1+\frac{3}{2}\right)^{2}}{\Gamma\left(1+\frac{3}{2} \right)}}}\) \\  & if \(0<|\tau_{3}|\leq\frac{1}{3}\) \\ \(\alpha=\exp\left(\begin{array}{c}-4.16202506-3.2616604018\cdot\ln(\tau_{3})-1.783702334\cdot\ln(\tau_{3})^{2}-0.770946644\cdot\ln(\tau_{3})^{3}-\\ 0.221698815\cdot\ln(\tau_{3})^{4}-0.041191426\cdot\ln(\tau_{3})^{5}-0.004665295 \cdot\ln(\tau_{3})^{6}-\\ 0.000287712\cdot\ln(\tau_{3})^{7}-0.000007207\cdot\ln(\tau_{3})^{8}\\ \end{array}\right)\) \\ Wilson-Hilferty & \(\frac{1}{3}<|\tau_{3}|\leq\frac{2}{3}\) \\ \(\alpha=\exp\left(\begin{array}{c}-5.264027693-9.134993593\cdot\ln(\tau_{3})-1 5.845477811\cdot\ln(\tau_{3})^{2}-19.874003352\cdot\ln(\tau_{3})^{3}-\\ 15.495267042\cdot\ln(\tau_{3})^{4}-6.752325319\cdot\ln(\tau_{3})^{5}-1.255615645 \cdot\ln(\tau_{3})^{6}\end{array}\right)\) \\  & \(\alpha=\frac{1}{3}<|\tau_{3}|<1\) \\ \(\alpha=\exp\left(\begin{array}{c}-7.712526023-93.06660109\cdot\ln(\tau_{3})-1.519099681\cdot\ln(\tau_{3})^{2}-1.602587644\cdot 10^{3}\cdot\ln(\tau_{3})^{4}-\\ 1.041173897\cdot 10^{5}\cdot\ln(\tau_{3})^{4}-4.152482998\cdot 10^{5}\cdot\ln(\tau_{3})^{5}-9.887282 \cdot 10^{5}\cdot\ln(\tau_{3})^{6}-\\ 1.287749298\cdot 10^{6}\cdot\ln(\tau_{3})^{7}-7.050767642\cdot 10^{5}\cdot\ln(\tau_{3})^{8} \end{array}\right)\) \\ \hline Pseudo-Weibull & \(\alpha=3.3784105-24.3298763\cdot\tau_{3}+1.3320721\cdot 10^{2}\cdot\tau_{3}^{2}-6.002157 \cdot\tau_{3}^{3}+2.0552631\cdot 10^{3}\cdot\tau_{3}^{4}-\\ 5.037878\cdot 10^{3}\cdot\tau_{3}^{5}+8.5230901\cdot 10^{3}\cdot\tau_{3}^{6}-9.6216631 \cdot 10^{3}\cdot\tau_{3}^{7}+6.8802849\cdot 10^{3}\cdot\tau_{3}^{8}-\\ 2.8078183\cdot 10^{3}\cdot\tau_{3}^{9}+4.9671387\cdot 10^{2}\cdot\tau_{3}^{10}\end{array}\) \\ \hline \(C_{s}=\frac{1}{0}\int\limits_{0}^{\left[\ln\left(\frac{s}{1-\frac{1}{1-\ln(s)- \tau_{3}|}}\right)-\psi\left(\frac{1}{2}\right)\cdot\Gamma\left(\frac{1}{2} \right)\cdot\Gamma\left(\frac{3}{2}\right)\right]}{\left[\int\limits_{0}^{1} \left[\ln\left(\frac{s}{1-\frac{1}{1-\ln(s)-\tau_{3}}}\right)^{2}dp-\left( \left(\frac{1}{2}+1\right)+\ln(\alpha)+\gamma_{s}\right)^{2}\right]^{15}}dp\) \\ \(\text{Generalized}\) & if \(0<\tau_{3}\leq 1/3\) \\ Gumbel & \(\alpha=-971.10886182\cdot 10^{-3}+6.3606944\cdot\tau_{3}-6.35405524\cdot\tau_{3}^{2}+15.035727 19\cdot\tau_{3}^{3}\) \\  & \(1/3<\tau_{3}<1\) \\ \(\alpha=\frac{-661.76303922\cdot 10^{-3}+4.3082328\cdot\tau_{3}-10.90504916\cdot\tau_{3}^{2}+37.12 88403\cdot\tau_{3}^{3}}{1-99.99892576\cdot 10^{-3}\cdot\tau_{3}}\) \\ \hline \end{tabular}