# Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions
## 5 Discussions
The PIII and EE distributions are an exception, but this is due to the fact that the variation of the shape parameter for the two estimation methods does not differ much. For example, Figure 5 shows the variation graph of the shape parameter for the Pearson III and Log-logistic distributions, for both estimation methods. Both skewness and L-skewness depend only on the shape coefficient \(\alpha\). It can be seen that the Pearson III distribution is more stable. A significant difference in the variation of the shape coefficient was registered for a narrow area, namely for \(\tau_{3}\) > 0.8, and \(C_{s}\) > 6.5, values that are not usually found in flood frequency analysis. In the case of the Log-logistic distribution, the variation of the parameter differs greatly for the two methods, over almost the entire range of values of skewness, and L-skewness, an aspect also observed in the results presented in the graph of Figure 4.  
Among the analyzed distributions with four or five parameters, including following the research carried out within the Faculty of Hydrotechnics, as well as the results obtained following the case study in this manuscript, the distributions from the Beta families recommended in flood frequency analysis using the L-moments method to estimate parameters, are the distributions KUM4, BEG, BG4, and BG5, which showed a stability related to both the length of the observed data and the presence of outliers. It can also be observed that these distributions exactly approximate all the indicators obtained based on linear moments. This criterion of calibration \(\tau_{3}\)-\(\tau_{4}\) is the most important criterion in the correct selection of distributions in analysis with L-moments. The results obtained with the RAE and RME performance indicators, both for MOM and for L-moments, provide relevant information  
Figure 5: The variation of parameter \(\alpha\). only in the area of the probabilities of the observed values; outside this area (left-hand, upper part of the graph) it loses its relevance. The manuscript does not exclude the applicability of other distributions from other families (Gamma, GEV, Pareto) in frequency analysis of extreme values, especially since these families were also analyzed within the research carried out in the Faculty of Hydrotechnics and presented in other materials [3; 19; 24]. In general, the L-moments method is a much more stable method than other estimation methods, being less influenced by the length of observed data as well as by the presence of extreme values (outliers), [8; 9; 17; 18]. However, in certain situations (small length of data, n \(<\) 20), the statistical indicators obtained with this method require some correction, which can be achieved using the least squares method [35]. In the Supplementary Materials, the graphs of the analyzed distributions are presented, highlighting the confidence intervals determined based on the simplified Chow relationship [11] for MOM, along with those based on the simplified relationship for L-moments presented in [19], both using the frequency factors specific to the distributions.