# Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions
## 1 Introduction  
An important part of the study of extreme events in hydrology involves frequency analysis. Hydrological frequency analysis is important because it determines extreme values with certain exceeding probabilities; these have a defining role in the design of dams and in water management [1, 2, 3, 4, 5]. Analyzing the international literature [6, 7, 8, 9, 10, 11, 12], the most frequently used distributions in flood frequency analysis are the distributions from the Gamma family (Pearson III, Generalized Gamma, Log Pearson), and the distributions from the GEV family (Weibull, Gumbel, Frechet). From the Beta distribution family, the most used distributions for flood frequency analysis are the Log-logistic distribution [11, 12], the Dagum distribution [13], the Burr distribution [14], the Kumaraswamy distribution [15]. In hydrological frequency analysis, to estimate the parameters of the distributions, the method of ordinary moments (MOM) and three-parameter distributions are used, as well as the method of linear moments (L-moments) using distributions chosen based on the L-skewness (\(\tau_{3}\))-L-kurtosis (\(\tau_{4}\)) variation criterion, recommended for distributions of at least four parameters. These parameter estimation methods are two of the most frequently used parameter estimation methods in hydrology, which is the reason for presenting only these two [3; 6; 11; 16]. In the case of MOM estimation, it is recommended to use three-parameter distributions because they can only calibrate moments up to order 3, namely the skewness that represents the \(m_{3}/m_{2}^{1.5}\) ratio. In the case of series smaller than 100 values, the skewness requires correction or can be chosen, as is the hydrological practice in Romania, depending on the origin of the extreme values [2; 3; 4; 5]. For moments of higher order, namely kurtosis (\(m_{4}/m_{2}^{2}\)), this cannot be corrected or chosen for small series, because the distributions would generate in some cases unrealistic values, especially in the area of small exceeding probabilities where there is no observed data [3; 17; 18; 19]. In the case of parameter estimation with L-moments, it is recommended to use distributions that have \(\tau_{3}\) and \(\tau_{4}\) natural values and that are very close to those of the observed data; distributions of at least four parameters are recommended, because the method requires the calibration of moments up to the fourth order. An advantage of using the L-moments method is that the method is more stable, being less affected by small data lengths, although in some cases it requires a certain correction. The correction of the statistical indicators obtained based on the L-moments method can be done using the least squares method. Taking into account the distributions from the Beta family applied in the literature regarding the determination of maximum flows [11; 12; 13; 14; 15], one of the objectives of this manuscript is the analysis of the applicability of other distributions belonging to the same family of distributions. Thus, this manuscript presents 14 statistical distributions of three, four and five parameters that are part of the families of Beta, Beta Prime and Beta Exponential distributions, namely the Generalized Beta distribution of five parameters (BG5), the Generalized Beta distribution of four parameters (BG4), the Kumaraswamy distribution (KUM4), the Pearson XII distribution (PXII), the Five-parameter Generalized Beta Prime distribution (BPG5), the Pearson VI distribution (PVI), the Lomax distribution (LMX), the Log-logistic distribution (LL), the Dagum distribution (DG), the four-parameter Burr distribution (BR4), the Paralogistic distribution (PR), the Inverse Paralogistic distribution (IPR), the Beta Exponential distribution (BE), and the Exponential Exponentiated (EE) distribution [11; 12; 13; 14; 15].