Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions

Abstract

This manuscript presents three families of distributions, namely the Beta, Beta Prime and Beta Exponential families of distributions. From all the distributions of these families, 14 statistical distributions of three, four and five parameters are presented that have applicability in the analysis of extreme phenomena in hydrology. These families of distributions were analyzed regarding the improvement of the existing legislation for the determination of extreme events, specifically the elaboration of a norm regarding frequency analysis in hydrology. To estimate the parameters of the analyzed distributions, the method of ordinary moments and the method of linear moments were used; the latter conforms to the current trend for estimating the parameters of statistical distributions. The main purpose of the manuscript was to identify other distributions from these three families with applicability in flood frequency analysis compared to the distributions already used in the literature from these families, such as the Log–logistic distribution, the Dagum distribution and the Kumaraswamy distribution. The manuscript does not exclude the applicability of other distributions from other families in the 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. All the necessary elements for their use are presented, including the probability density functions, the complementary cumulative distribution functions, the quantile functions and the exact and approximate relations for estimating parameters. A flood frequency analysis case study was carried out for the Prigor RiverRiver, to numerically present the proposed distributions. The performance of this distributions were evaluated using the relative mean error, the relative absolute error and the L-skewness–L-kurtosis diagram. The best fit distributions are the Kumaraswamy, the Generalized Beta Exponential and the Generalized Beta distributions, which presented a stability related to both the length of the data and the presence of outliers.

Predicting Flood Frequency with the LH-Moments Method: A Case Study of Prigor River, Romania

“…All this research is part of much more complex scientific research carried out within the Faculty of Hydrotechnics, in which a large number of distributions and families of distributions were analyzed using the method of ordinary moments and the method of linear moments, with results concretized in other papers [18][19][20][21]27,30].…”

Revisiting the use of the Gumbel distribution: A Frequency Analysis Perspective

“…In the case of the L-moments method, there are clear criteria for selecting the best model, namely the calibration of the indicator values of L-skewness ( 3 ) and L-kurtozis ( 4 ) of the observed data [2,[65][66][67]69,[71][72][73][87][88][89][90][91][92][93][94]. Unfortunately, the Gumbel distribution is not defined, like the twoparameter Log-normal or Gamma distribution, by a variation curve of these indicators, but has constant values regardless of the observed data analyzed [91].…”

Predicting Future Flood Risks in the Face of Climate Change: A Frequency Analysis Perspective

“…In this section, only the inverse functions of the analyzed distributions are presented (see Table 2), the L-and LH-moments are based on the inverse function. Density functions and cumulative functions can be found in others materials [13,32,33].…”

“…Considering that these inverse functions can be expressed using frequency factors [11,13], the exact and approximate relationships of these frequency factors are presented in the Supplementary File.…”

“…The Gamma, the Generalized Pareto, and the GEV family distributions are some of the probability distributions that are most frequently employed in FFA or regional FFA [1,[3][4][5][6][7][8]. But, in recent materials [9][10][11][12][13], other distributions and families of distributions have been introduced in FFA, such as distributions from the generalized beta family, generalized beta prime, and beta exponential [13], using the method of ordinary moments (MOM) and the L-moments method as parameter estimation.…”