# Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions
## 1 Introduction
The estimation methods of the parameters of these distributions are MOM and L-moments, with the latter being necessary to solve some nonlinear systems of equations, which leads to some difficulties in using these distributions. Thus, for the ease of application, parameter approximation relations are presented for some of these distributions using polynomial [20], exponential or rational functions. All the mathematical elements necessary to use these distributions in the analysis of extreme events, especially in flood frequency analysis, are presented here. In this manuscript, new elements such as the expressions of the cumulative complementary functions; the inverse functions for BG5, BG4, PXII, BPG5, PVI, LMX, BE, EE; the approximation relations for parameter estimation for PXII, PR, IPR, EE; and the relations for parameter estimation with MOM for LMX, PR, IPR, facilitate the ease of using these distributions in flood frequency analysis. Some of the quantiles of the analyzed distributions do not have explicit forms; they are represented in this manuscript with the help of the predefined function from Mathcad, which is equivalent to other functions from other dedicated programs (e.g., the Beta.Inv function from Excel, etc.). Thus, all these new elements for these distributions presented in Table 1 will help hydrology researchers to use these distributions easily. This is the first time that the BG5, BPG5, LMX, PR, IPR, BE and EE distributions are used in flood frequency analysis. The raw and central moments of the analyzed distributions were determined using the methodology presented in the Supplementary Materials, based on the probability density functions. In the case of estimation with L-moments, the determination of L-moments uses the substitution methodology (variable change) using the expression of the inverse function. This parameter estimation method is based exclusively on the inverse function of the distribution; the presentation of the expressions of the inverse functions of some distributions that have not been presented so far are novel elements, and are essential in the application of distributions using the L-moments method. In order to verify the performances of the proposed distributions, a flood frequency analysis was carried out, using the Prigor River as a case study. All results are presented in comparison with the Pearson III distribution, which is the parent distribution in flood frequency analysis in Romania [2; 3; 4; 5; 21]. Comparing the results and choosing the best distribution was based on the performance indicators: relative mean error (RME) [22], relative absolute error (RAE) [22] and L-skewness-L-kurtosis diagrams [3; 8; 11]. The manuscript is organized as follows: The description of the statistical distributions by presenting the density function, the complementary cumulative function and the quantile function, is given in Section 2.1. The presentation of the relations for exact calculation and the approximate relations for determining the parameters of the distributions is given in Section 2.2. A case study applying these distributions to flood frequency analysis for the Prigor River is presented in Section 3. Results, discussions and conclusions are presented in Sections 3-5, respectively.