# Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions
## 5 Discussions  
Flood frequency analysis is necessary to determine the maximum flows with certain exceeding probabilities necessary for the design of hydrotechnical constructions and establishing the bankfull discharge. The choice of distributions and the methods of estimating the parameters of these distributions have an important role in the correct performance of such an analysis. In Romania, the regulations [34] do not provide sufficient rigorous mathematical criteria. The normative approach is deficient, analyzing in a random way distributions with two or three parameters, and only the method of ordinary moments is treated as the method of parameter estimation. Thus, taking into account the modern approaches of using the L-moments method in estimating the parameters of the distributions in the frequency analysis of extreme events, this manuscript presents 14 statistical distributions, from three families of Beta distributions, with applicability in flood frequency analysis, which use the method of ordinary moments and L-moments for parameter estimation. An important criterion for choosing the distributions in the analysis with L-moments is \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Distr.**} & \multicolumn{6}{c}{**Statistical Measures**} \\ \cline{2-7}  & \multicolumn{6}{c}{**Methods of Parameter Estimation**} & \multirow{2}{*}{**Observed Data**} \\ \cline{2-7}  & \multicolumn{2}{c}{**MOM**} & \multicolumn{2}{c}{**L-Moments**} & \\ \cline{2-7}  & **RME** & **RAE** & **RME** & **RAE** & \(\tau_{3}\) & \(\tau_{4}\) & \(\tau_{3}\) & \(\tau_{4}\) \\ \hline PE3 & 0.0231 & 0.0841 & 0.0219 & 0.0885 & & 0.192 & & \\ \hline BG5 & 0.0186 & 0.0871 & 0.0165 & 0.0712 & & 0.228 & & \\ \hline BG4 & 0.0432 & 0.1198 & 0.0237 & 0.0912 & & 0.228 & & \\ \hline KUM4 & 0.0214 & 0.0804 & 0.022 & 0.0843 & & 0.228 & & \\ \hline PXII & 0.0660 & 0.2092 & 0.0329 & 0.1338 & & 0.118 & & \\ \hline BPG5 & 0.0742 & 0.2481 & 0.0171 & 0.0734 & & 0.228 & & \\ \hline PVI & 0.0233 & 0.1098 & 0.0149 & 0.0639 & & 0.271 & & \\ \hline LMX & 0.0184 & 0.0784 & 0.0181 & 0.0765 & & 0.221 & 0.399 & 0.228 \\ \hline LL & 0.0537 & 0.2060 & 0.0165 & 0.0715 & & 0.299 & & \\ \hline DG & 0.0480 & 0.2141 & 0.0164 & 0.0714 & & 0.300 & & \\ \hline BR4 & 0.0330 & 0.1424 & 0.0214 & 0.0910 & & 0.228 & & \\ \hline PR & 0.0398 & 0.1696 & 0.0155 & 0.0672 & & 0.272 & & \\ \hline IPR & 0.0357 & 0.1506 & 0.0158 & 0.0678 & & 0.295 & \\ \hline BEG & 0.0441 & 0.1399 & 0.0237 & 0.0912 & & 0.228 & \\ \hline EE & 0.0250 & 0.0872 & 0.0224 & 0.0907 & & 0.188 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Distributions performance values. the \(\tau_{3}\)-\(\tau_{4}\) variation; it is recommended to use distributions that have the values of these indicators very close to those of the observed data. Figure 11 from Appendix A shows the graph and \(\tau_{3}\)-\(\tau_{4}\) variation relations for some of the most used distributions in hydrology, including some of the distributions analyzed in this manuscript. All the results obtained in the case study are presented compared to the Pearson III distribution, which is considered the parent distribution in Romania. From the obtained results it can be seen that for the distributions with three parameters, the calibration is satisfactory for MOM, the skewness being chosen according to the genesis of the flows, as is the hydrological practice in Romania. It should be mentioned that for the MOM estimation, both for the three-parameter distributions and for the four- and five-parameter distributions, the resulting values are characterized by a degree of uncertainty due to the fact that a proper calibration of the kurtosis cannot be done. For the L-moments method, the results obtained using the three-parameter distributions are generally unsatisfactory, generating unrealistic values in the area of small exceedance probabilities (left hand), and do not achieve a proper calibration of the high-order linear moments.