# Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions
## 2 Methods
#### 2.2.12 Inverse Paralogistic (IPR)  
The equations needed to estimate the parameters with MOM have the following expressions:  
\[\mu=\gamma+\beta\cdot\frac{\Gamma\left(\alpha+\frac{1}{\alpha}\right)\cdot \Gamma\left(1-\frac{1}{\alpha}\right)}{\Gamma\left(\alpha\right)} \tag{86}\]  
\[\sigma^{2}=\frac{\beta^{2}}{\Gamma\left(\alpha\right)}\cdot\left(\Gamma \left(1-\frac{2}{\alpha}\right)\cdot\Gamma\left(\alpha+\frac{2}{\alpha} \right)-\frac{\Gamma\left(1-\frac{1}{\alpha}\right)^{2}\cdot\Gamma\left( \alpha+\frac{1}{\alpha}\right)^{2}}{\Gamma\left(\alpha\right)}\right) \tag{87}\]  
Because it is too long, the relationship for estimating skewness is presented in Appendix C.  
The parameter \(\alpha\) can be obtained approximately depending on the skewness coefficient, using the following rational functions (\(1.3\leq C_{s}\leq 7\)):  
\[\alpha=\exp\left(\begin{array}{l}9.1148085-48.5411515\cdot\ln(C_{s})+178.42 82739\cdot\ln(C_{s})^{2}-411.0985176\cdot\ln(C_{s})^{3}+\\ 611.4661111\cdot\ln(C_{s})^{4}-596.305336\cdot\ln(C_{s})^{5}+378.4065467\cdot \ln(C_{s})^{6}-\\ 150.2898201\cdot\ln(C_{s})^{7}+33.8871674\cdot\ln(C_{s})^{9}-3.3074833\cdot\ln( C_{s})^{9}\end{array}\right) \tag{88}\]  
\[\beta=\frac{\sigma}{\sqrt{\frac{1}{\Gamma\left(\alpha\right)}\cdot\left( \Gamma\left(1-\frac{2}{\alpha}\right)\cdot\Gamma\left(\alpha+\frac{2}{\alpha }\right)-\frac{\Gamma\left(1-\frac{1}{\alpha}\right)^{2}\cdot\Gamma\left( \alpha+\frac{1}{\alpha}\right)^{2}}{\Gamma\left(\alpha\right)}\right)}} \tag{89}\]  
\[\gamma=\mu-\beta\cdot\frac{\Gamma\left(1-\frac{1}{\alpha}\right)\cdot\Gamma \left(\alpha+\frac{1}{\alpha}\right)}{\Gamma\left(\alpha\right)} \tag{90}\]  
The equations needed to estimate the parameters with L-moments have the following expressions:  
\[L_{1}=\gamma+\frac{\beta\cdot\Gamma\left(\frac{1}{\alpha}+\alpha\right)\cdot \Gamma\left(1-\frac{1}{\alpha}\right)}{\Gamma\left(\alpha\right)} \tag{91}\]\[L_{2}=\beta\cdot\Gamma\bigg{(}1-\frac{1}{\alpha}\bigg{)}\cdot\left(\frac{\Gamma \Big{(}\frac{1}{\alpha}+2\cdot\alpha\Big{)}}{\Gamma(2\cdot\alpha)}-\frac{\Gamma \Big{(}\frac{1}{\alpha}+\alpha\Big{)}}{\Gamma(\alpha)}\right) \tag{92}\]  
\[L_{3}=\beta\cdot\Gamma\bigg{(}1-\frac{1}{\alpha}\bigg{)}\cdot\left(\frac{\Gamma \Big{(}\alpha+\frac{1}{\alpha}\Big{)}}{\Gamma(\alpha)}+\frac{2\cdot\Gamma \Big{(}3\cdot\alpha+\frac{1}{\alpha}\Big{)}}{\Gamma(3\cdot\alpha)}-\frac{3 \cdot\Gamma\Big{(}2\cdot\alpha+\frac{1}{\alpha}\Big{)}}{\Gamma(2\cdot\alpha)}\right) \tag{93}\]  
An approximate form can be adopted based on the parameter estimation depending on L-skewness, as follows:  
if \(0.17\leq\tau_{\mathrm{s}}<1/3\):  
\[\alpha=\frac{5.07675\cdot 10^{3}-7.55684\cdot 10^{4}\cdot\tau_{3}+5.36797\cdot 1 0^{5}\cdot\tau_{3}^{2}-1.35970\cdot 10^{6}\cdot\tau_{3}^{3}+1.30233\cdot 10^{6} \cdot\tau_{3}^{4}}{1-4.59437\cdot 10^{3}\cdot\tau_{3}+2.70167\cdot 10^{4} \cdot\tau_{3}^{2}} \tag{94}\]  
if \(1/3<\tau_{\mathrm{s}}<1\):  
\[\begin{array}{c}\alpha=5.37952\cdot 10-4.84016\cdot 10^{2}\cdot\tau_{3}+2.02 215\cdot 10^{3}\cdot\tau_{3}^{2}-4.79644\cdot 10^{3}\cdot\tau_{3}^{3}+\\ 6.86370\cdot 10^{3}\cdot\tau_{3}^{4}-5.88301\cdot 10^{3}\cdot\tau_{3}^{5}+2.78592 \cdot 10^{3}\cdot\tau_{3}^{5}-5.61108\cdot 10^{2}\cdot\tau_{3}^{7}\end{array} \tag{95}\]  
\[\beta=\frac{L_{2}}{\Gamma\Big{(}1-\frac{1}{\alpha}\Big{)}\cdot\left(\frac{ \Gamma\big{(}\frac{1}{\alpha}+2\cdot\alpha\big{)}}{\Gamma(2\cdot\alpha)}- \frac{\Gamma\big{(}\frac{1}{\alpha}+\alpha\big{)}}{\Gamma(\alpha)}\right)} \tag{96}\]  
\[\gamma=L_{1}-\frac{\beta\cdot\Gamma\Big{(}\frac{1}{\alpha}+\alpha\Big{)}\cdot \Gamma\Big{(}1-\frac{1}{\alpha}\Big{)}}{\Gamma(\alpha)} \tag{97}\]