Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events

Abstract

The manuscript presents the applicability of the Gumbel distribution in the frequency analysis of extreme events in hydrology. The advantages and disadvantages of using the distribution are highlighted, as well as recommendations regarding its proper use. A literature review was also carried out regarding the methods for estimating the parameters of the Gumbel distribution in hydrology. Thus, for the verification of the methods, case studies are presented regarding the determination of the maximum annual flows and precipitations using nine methods for estimating the distribution parameters. The influence of the variability of the observed data lengths on the estimation of the statistical indicators, the estimation of the parameters, and the quantiles corresponding to the field of small exceedance probabilities (p < 1%) is also highlighted. In each case, the results are analyzed compared to those obtained with the Generalized Extreme Value distribution, the four-parameter Burr distribution, and the five-parameter Wakeby distribution estimated using the L-moments method. The results of the case studies highlight and reaffirm the statistical, mathematical, and hydrological recommendations regarding the avoidance of applying the Gumbel distribution in flood frequency analysis and its use with reservations in the case of maximum precipitation analysis, especially when the statistical indicators of the analyzed data are not close to the characteristic ones and unique to the distribution.

1. Introduction

Frequency analysis represents a common statistical method used to assess the likelihood and magnitude of events. The primary objective of frequency analysis is to estimate the probability of different event magnitudes occurring over a specified time period. In hydrology, frequency analysis is a direct way of determining the magnitude and return periods of extreme events, such as floods, droughts, and extreme rainfall.

These types of analyses are based exclusively on the use of statistical distributions, such as Pearson III, Log-Pearson III, Log-normal, Weibull, etc., whose parameters are estimated using different parameter-estimation methods (the method of ordinary moments, the linear moments method, the least squares method, the mixed-moment method, the maximum likelihood method, etc.) [1].

These forecasted values are of particular importance in flood frequency analysis (FFA), as they represent the maximum flows necessary for the design of dam-type hydrotechnical constructions related to annual exceeding probabilities (0.01%, 0.1%) considered “impossible floods’’. These are “large events, or different types of events, that exceed the expectations based on historical experience. Such impossible events include floods whose probability of occurrence is considered too small to act, but also events that go beyond any imagination” [2]. Determining them accurately leads to the avoidance of additional costs, which, in some cases, are significant (leading to the over-dimensioning of protection works), the avoidance of loss of human life, and the avoidance of important damages due to the under-dimensioning of the design flows that can generate a failure.

One of the most popular statistical distributions is the Gumbel distribution. The Gumbel distribution (a particular case of the Generalized Extreme Value) was and remains one of the most commonly used statistical distributions in the frequency analysis of extreme events in hydrology. This aspect is mainly due to the simple parameter estimation expressions, as well as the simple and accessible expression of the quantile function (inverse function). In hydrology, the applicability of the Gumbel distribution is diverse, being mainly used for frequency analysis of maximum flows [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44], maximum precipitation, and the construction of IDF curves [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60], minimum flows [61,62,63,64,65,66], etc.

The main and only advantage of the Gumbel distribution is the simplicity and accessibility of expressions and relationships. Regarding the disadvantages, we can state as the main disadvantage the limited flexibility of this distribution in modeling various forms of skewness, which generally leads to the limitation of the application of this distribution. It is a distribution whose statistical indicators have constant values, namely skewness $C_s=1.1401$ and kurtosis $C_k=5.398$ for the method of ordinary moments (MOM) [3,4,5,6], L-skewness ${\tau }_3=0.17$ and L-kurtosis ${\tau }_4=0.15$ for the method of linear moments (L-moments) [3,4,5,6,67,68,69,70], LH-skewness ${\tau }_{H3}=0.243$ and LH-kurtosis ${\tau }_{H4}=0.147$ for the method of high-order linear moments (first-level LH1-moments), and ${\tau }_{H3}=0.272$ and ${\tau }_{H4}=0.149$ for the method of high-order linear moments (second-level LH2-moments). Another major disadvantage is that for certain values of the coefficient of variation ($C_v>0.60946$), the values generated in the field of high-exceeding probabilities are negative. Thus, for a coefficient of variation $C_v=1$, the distribution generates negative values in the range of 87–99.9%, for $C_v=1.5$, negative values appear in the range of 75–99.9%, and for $C_v=2$, negative values are found in the range of 66–99.9%.

Regarding the use of the Gumbel distribution with different parameter estimation methods, it is applied using the method of ordinary moments (MOM) [3,4,5,6,7], the method of linear moments (L-moments) [4,66,67,68,69,70], the method of linear moments high-order (LH-moments) [71,72,73,74,75], the weighted moments method (PWM) [4,76,77], the maximum likelihood method (MLE) [4,76,77], the least squares method (LSM) [4,76,78,79], the weighted least squares method (WLSM) [79], the entropy method (ENT) [78,80], and the mixed moments method [4,76,78].

Thus, the main objective of this manuscript is to analyze the limited applicability of the Gumbel distribution in frequency analysis in hydrology in different particular cases with different morphological, statistical, and hydrological characteristics. Three rivers related to the territory of Romania are analyzed (through flood frequency analysis) using the series of maximum annual flows (AMS) with different morphometric characteristics (watershed areas, average slopes, average altitudes, sinuosity, etc.) and statistical and hydrological characteristics (available data lengths, coefficient of variation, skewness, and kurtosis). Regarding the determination of the maximum annual precipitation, the performances of the Gumbel distribution are analyzed in two case studies with different statistical, mathematical, and hydrological peculiarities.

The differences and the behaviors of the curves of the inverse distribution function, estimated with the nine previously mentioned methods, are highlighted. The Bayesian approach is not treated in this manuscript because although important contributions have been made, it remains a method with limitations, being strongly influenced by subjective aspects in the choice of the a priori distribution, requiring complex analyses and strong hypotheses for its choice [81,82,83,84].

In the case of flood frequency analysis (FFA), it is important to mention that it “…… may result in significant jumps in the estimates of design (flood) quantiles along with the lengthening series of maximum flows” [85]. The biases of the quantiles of the distribution (sampling analysis) due to the variability of the analyzed data lengths for two of the most used parameter estimation methods are presented: MOM and L-moments. These also represent the main methods for parameter estimation in the regionalization analyses of extreme events because they are characterized by statistical indicators (arithmetic mean, mean square deviation, skewness, and kurtosis) that can be determined regionally.

The results of the case studies are presented compared to those obtained with three other much more flexible distributions, such as the GEV distribution and the four- and five-parameter Burr/Wakeby distributions, with parameters estimated using the L-moments method. The choice of the L-moments method as a reference is due to the main advantages presented and highlighted in many scientific materials [67,70,71,72,86,87,88,89,90,91,92]. It is robust in estimating statistical indicators and in estimating the parameters and quantiles of distributions of variable lengths of data, as well as the presence of outlier values. Also, the method presents rigorous criteria for choosing the best model, namely compliance with the condition that the higher-order statistical indicators (L-skewness and L-kurtosis) specific to the chosen distribution come as close as possible to the corresponding ones of the analyzed data [4,70,73,74,89,90,91,92,93,94]. The comparison with the values generated by the Burr/Wakeby distribution was based on the consideration that those methods, being characterized by a high number of parameters, manage the exact calibration of all four linear moments, the results obtained being characterized by a high degree of confidence. In previous materials [93], it was demonstrated that by choosing the appropriate distribution of two parameters and by observing the specific criteria of the L-moments method, the errors in the estimation of quantiles can be greatly reduced. If it is desired to use a two-parameter distribution, it is recommended to use the L-moments method, and the preselection of the distributions and the selection of the model distribution should be performed according to the criteria specific to the method, namely according to the values and variation curves (diagrams of variation) of the L-skewness and L-kurtosis indicators. In the latest period, numerous relations and diagrams of variations have been developed and presented for a significant number of two- and three-parameter distributions [73,74,75,89,90,91,92,93,94,95,96].

Considering that by using the frequency analysis we want to forecast the values corresponding to rare and very rare events (annual exceedance probability p < 1%) characteristic of long return periods (average return period T > 100 years), the performance of the Gumbel distribution is analyzed on two distinct levels of the maximum annual probabilities of non-exceedance, namely on the interval between p_1% and p_100%, in the case of maximum precipitation (h_p), where data are generally available and generally the value of the quantile with an average return period of 200 years is of interest, and for the interval p_0.01%~p_1%, in the case of maximum flows (Q_p), where most of the time there are no data available, representing the interval in which the extrapolation analysis of the available data is desired.

Among the most important new contributions made in this manuscript, regarding the Gumbel distribution and the methods of using it in the analysis of extreme events, I mention the following: (1) presentation of the relationships for estimating the parameters and frequency factors of the Gumbel distribution using the first- and second-order LH-moments method; (2) presentation of the systematic biases of the Gumbel distribution (for both parameters and quantiles) depending on the statistical variability of extreme events (statistical indicators, available string lengths) for the most common parameter estimation methods; (3) a comparative analysis regarding the behavior of the Gumbel distribution for a significant number of parameter estimation methods; and (4) establishing clear and rigorous recommendations regarding the applicability and usefulness of the Gumbel distribution.

This is the first time that a comparative analysis is performed regarding the applicability of the Gumbel distribution using so many parameter estimation methods. This information and the obtained results will help researchers to use this statistical distribution appropriately and rigorously.

2. Methods

This section presents in detail the statistical and mathematical elements necessary to apply the Gumbel distribution using the nine methods of estimating the analyzed parameters. The variation graphs of the inverse function related to the variability of the available data lengths and the data variability (the theoretical choice of the usual $C_v$ in FFA) are also presented, highlighting the systematic biases of the distribution for the annual maximum values (with MOM and L-moments) for the annual exceedance probabilities of interest: 1%, 0.1%, and 0.01% [3,4,73,74,75].

This analysis of the evidence of the systematic biases of the distribution follows the stage of choosing the best model. In the second stage, it is considered that the analyzed values (the observed data) are drawn from a Gumbel distribution.

Figure 1 shows the flowchart of the presented methodology.

2.1. Estimating the Parameters of the Gumbel Distribution

The Gumbel distribution is a particular case of the Generalized Extreme Value (GEV) distribution where the shape parameter $\alpha =0$ [3,4,5,6]. It is also known as the type I extreme value distribution, the LogWeibull distribution, the Fisher–Tippett type I distribution, or the Gompertz distribution [97].

The probability density function, the complementary cumulative function, and the quantile function of the Gumbel distribution have the following expressions [3,4,5,6,66,67,68,69,76]:

-

Probability density function (pdf):

$$f\left(x\right)={\displaystyle \frac{1}{\beta }}\cdot \exp \left(-{\displaystyle \frac{x-\gamma }{\beta }}-\exp \left(-{\displaystyle \frac{x-\gamma }{\beta }}\right)\right)$$

(1)

-

The cumulative distribution function (cdf):

$$F\left(x\right)=1-\exp \left(-\exp \left({\displaystyle \frac{x-\gamma }{\beta }}\right)\right)$$

(2)

-

The quantile function:

$$x\left(p\right)=\gamma -\beta \cdot \ln \left(-\ln \left(1-p\right)\right)$$

(3)

where $\beta >0$ is the scale parameter and $-\infty <\gamma <x$ is the position parameter; $-\infty <x<\infty$.

The quantile function can also be expressed, for MOM and L- and LH-moments, using the frequency factor $K\left(p\right)$, as follows:

-

MOM:

$$x\left(p\right)=\mu +\sigma \cdot K\left(p\right)$$

(4)

-

L-moments:

$$x\left(p\right)=L_1+L_2\cdot K\left(p\right)$$

(5)

-

LH-moments:

$$x\left(p\right)=L_{H1}+L_{H2}\cdot K\left(p\right)$$

(6)

where $\mu$ is the arithmetic mean (expected value), ${\sigma }^2$ represents the variation, and ${\gamma }_e=0.577216$ is Euler’s constant; $L_1$ and $L_2$ represent the first two linear moments; and $L_{H1}$ and $L_{H2}$ represent the first two high-order linear moments.

2.1.1. Method of Ordinary Moments (MOM)

The conditions and expressions of the parameters of the Gumbel distribution using MOM are as follows [3,4,69]:

$$\mu =\gamma +\beta \cdot {\gamma }_e$$

(7)

$${\sigma }^2={\displaystyle \frac{{\beta }^2\cdot {\pi }^2}{6}}$$

(8)

The two expressions of the distribution parameters result from Equations (7) and (8):

$$\gamma =\mu -\beta \cdot {\gamma }_e$$

(9)

$$\beta ={\displaystyle \frac{\sqrt{6}\cdot \sigma }{\pi }}$$

(10)

The frequency factor of the distribution using the MOM method is defined [3,88]:

$$K\left(p\right)=-{\displaystyle \frac{\sqrt{6}}{\pi }}\cdot \left(\ln \left(-\ln \left(1-p\right)\right)+{\gamma }_e\right)$$

(11)

2.1.2. The Method of Linear Moments (L-Moments)

To determine the expressions of the parameters using the L-moments method, it is necessary to impose the following two conditions [4,69,76,89]:

$$L_1=\gamma +\beta \cdot {\gamma }_e$$

(12)

$$L_2=\beta \cdot \ln \left(2\right)$$

(13)

The expressions of the two parameters are the following:

$$\gamma =L_1-\beta \cdot {\gamma }_e$$

(14)

$$\beta ={\displaystyle \frac{L_2}{\ln \left(2\right)}}$$

(15)

The frequency factor of the distribution using the L-moments method has the following explicit form:

$$K\left(p\right)={\displaystyle \frac{-\ln \left(-\ln \left(1-p\right)\right)-{\gamma }_e}{\ln \left(2\right)}}$$

(16)

2.1.3. The Method of High-Order Linear Moments (LH-Moments)

The equations of the conditions necessary to estimate the parameters using the method of linear moments of order (first-level LH1-moments) are the following:

$$L_{H1}=\gamma +\beta \cdot \left(\ln \left(2\right)+{\gamma }_e\right)$$

(17)

$$L_{H2}=-\beta \cdot {\displaystyle \frac{3}{2}}\cdot \ln \left({\displaystyle \frac{2}{3}}\right)$$

(18)

The position and scale parameters have the following expressions:

$$\gamma =L_{H1}-\beta \cdot \left({\gamma }_e+\ln \left(2\right)\right)$$

(19)

$$\beta ={\displaystyle \frac{-2\cdot L_{H2}}{3\cdot \ln \left(\frac{2}{3}\right)}}$$

(20)

The frequency factor of the distribution using the LH1-moments is

$$K\left(p\right)={\displaystyle \frac{2\cdot \ln \left(-\ln \left(1-p\right)\right)+\ln \left(4\right)+2\cdot {\gamma }_e}{\ln \left(\frac{8}{27}\right)}}$$

(21)

In the case of the method of second-level linear moments, the two conditions are

$$L_{H1}=\gamma +\beta \cdot \left(\ln \left(3\right)+{\gamma }_e\right)$$

(22)

$$L_{H2}=-\beta \cdot 2\cdot \ln \left({\displaystyle \frac{3}{4}}\right)$$

(23)

It follows from Equations (22) and (23) that

$$\gamma =L_{H1}-\beta \cdot \left({\gamma }_e+\ln \left(3\right)\right)$$

(24)

$$\beta ={\displaystyle \frac{-L_{H2}}{2\cdot \ln \left(\frac{3}{4}\right)}}$$

(25)

The frequency factor of the distribution using the LH2-moments is

$$K\left(p\right)={\displaystyle \frac{\ln \left(-\ln \left(1-p\right)\right)+\ln \left(3\right)+{\gamma }_e}{2\cdot \ln \left(\frac{3}{4}\right)}}$$

(26)

2.1.4. The Probability Weighted Moment Method (PWM)

The general equation for obtaining the weighted moments has the following mathematical expression [4,76,77]:

$$W_{k+1}={\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n\left(X_i\cdot {\left(1-P_i\right)}^k\right)}$$

(27)

where $n$ represents the length of the analyzed data string; $X$ is the string of recorded values in ascending order; $P_i$ is the chosen empirical probability; and $k=0,1,2,\dots$. For $k=0$, we obtain the expected value.

Like the least squares method (LSM), this also has an important subjective component based on choosing the empirical probability.

The parameter equations are

$$\beta ={\displaystyle \frac{W_1-2\cdot W_2}{\ln \left(2\right)}}$$

(28)

$$\gamma =W_1-{\gamma }_e\cdot \beta$$

(29)

2.1.5. The Method of Maximum Likelihood Estimation (MLE)

The relations for estimating parameters with MLE are the following [4,76,78,79]:

$$\beta ={\displaystyle \frac{\mu \cdot {\displaystyle \sum _{i=1}^n\exp \left(-X_i\cdot \frac{1}{\beta }\right)}-{\displaystyle \sum _{i=1}^n{X}_i\cdot \exp \left(-X_i\cdot \frac{1}{\beta }\right)}}{{\displaystyle \sum _{i=1}^n\exp \left(-X_i\cdot \frac{1}{\beta }\right)}}}$$

(30)

$$\gamma =\beta \cdot \ln \left({\displaystyle \frac{n}{{\displaystyle \sum _{i=1}^n\exp \left(-X_i\cdot \frac{1}{\beta }\right)}}}\right)$$

(31)

2.1.6. The Least Squares Method (LSM)

To determine the parameters using the LSM method, it is necessary to solve the following system of nonlinear equations [78]. The relations of the system necessary for the estimation of the parameters are presented in Appendix A.

2.1.7. The Weighted Least Squares Method (WLSM)

The nonlinear equations of the system needed to estimate the parameters with WLSM are the following [78]. The relations of the system are presented in Appendix A.

2.1.8. The Entropy Weight Method (ENT)

The scale parameter is obtained by finding the solution of the following nonlinear equation [78]:

$$\beta ={\displaystyle \frac{\mu }{{\gamma }_e+\ln \left({\displaystyle \sum _{i=1}^n\exp \left(-X_i\cdot \frac{1}{\beta }\right)}\right)}}$$

(32)

With the scale parameter thus known, the position parameter is determined using Equation (31).

2.1.9. The Method of MIXED Moments (MIX)

The equations for determining the parameters using the MIX method have the following expressions [78]:

$$\beta ={\displaystyle \frac{\sigma }{1.28255}}$$

(33)

$$\gamma =\beta \cdot \ln \left(1+{\displaystyle \frac{\mu }{\beta }}+{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{X}_i^2}}{2\cdot n\cdot {\beta }^2}}\right)$$

(34)

Regarding the estimation of the parameters of the four-parameter Burr and five-parameter Wakeby distributions using the L-moments method, they are presented in previous materials [93,95].

2.2. Systematic Biases Due to the Variability of Data Length

This section presents, for the Gumbel distribution, a comparative analysis of the behavior of the quantile function (in the field of low exceedance probabilities, events with a long return period), depending on the number of values ($n$) and depending on the coefficient of variation ($C_v$), for different empirical probabilities ($P_{empiric}$) chosen in the sampling. The biases of the distribution parameters are highlighted, as well as the biases of the values generated for four small annual non-exceedance probabilities (p = 0.01%, 0.1%, 1%, and 5%), compared to the theoretical values (n > 1000 values) characteristic of the Gumbel distribution.

The analysis is performed through sampling using the inverse function (with parameters estimated for the theoretical values, n > 1000) and Hazen empirical probability. The choice of the best empirical probability for sampling was made so that the differences between the generated values and the theoretical ones are minimal. This must be established depending on the parameter estimation method and the analyzed distribution [98]. The influence of the choice of the empirical probability is thus presented, with aspects that also influence the estimation of the parameters using the LSM or PWM methods. A number of 13 empirical probabilities were analyzed [4,98], namely Blom, Weibull, Beard, Hazen, Cunnane, Gringorten, Adamowski, Chegodayev, Filliben, Hirsch, IEC56, McClung and Mears, and Landwehr/APL.

2.2.1. Method of Ordinary Moments

Being a two-parameter distribution, the MOM analysis is based on the theoretical values of the coefficient of variation ($C_v$). In general, in frequency analysis in hydrology, it takes values between 0.1 and 5. Regarding the sampling (both for MOM and for L-moments), this is performed for a number of values of $n=15,25,80$, lengths generally available in real data applications. In Figure 2 are presented the graphs of the behavior of the inverse function for different values of the coefficient of variation and according to the empirical probabilities that generated the largest biases (Weibull) and the smallest (Hazen).

The theoretical bias values in estimating the parameters of the Gumbel distribution, for all of the analyzed values (data lengths and coefficient of variation values), are presented, as a percentage, in

Table 1. The theoretical biases for the scale parameter: MOM.

The Scale Parameter $\mathit{\beta }$
The Coefficient of Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Theoretical Bias [%]
$C_v=0.1$	6.33	1.07	13.15	2.57	17.63	3.69
$C_v=0.6$	6.33	1.07	13.15	2.57	17.63	3.69
$C_v=1$	6.33	1.07	13.15	2.57	17.63	3.69
$C_v=2$	6.33	1.07	13.15	2.57	17.63	3.69
$C_v=4$	6.33	1.07	13.15	2.57	17.63	3.69

and

Table 2. The theoretical biases for the position parameter: MOM.

The Scale Parameter $\mathit{\gamma }$
The Coefficient of Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Theoretical Bias [%]
$C_v=0.1$	−0.13	−0.02	−0.24	−0.03	−0.31	−0.02
$C_v=0.6$	−1.04	−0.16	−1.89	−0.23	−2.4	−0.19
$C_v=1$	−2.3	−0.36	−4.19	−0.5	−5.3	−0.41
$C_v=2$	−25.31	−3.93	−46.09	−5.51	−58.39	−4.53
$C_v=4$	6.32	0.98	11.51	1.37	14.58	1.13

.

An interesting aspect was highlighted by Gaume [81], namely the transmission of the parameter biases in the quantile biases, this being influenced by the parameter estimation functions specific to the MOM and L-moments method.

Thus,

Table 3. The theoretical biases for rare events (p = 0.01%, 0.1%, 1% and 5%): MOM.

Q0.01%
The Coefficient of Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Theoretical Bias [%]
$C_v=0.1$	2.63	0.42	5.5	1.07	7.4	1.55
$C_v=0.6$	5.26	0.89	11	2.16	14.75	3.14
$C_v=1$	5.72	0.97	11.9	2.35	16	3.4
$C_v=2$	6.11	1	12.74	2.51	17.1	3.63
$C_v=4$	6.33	1.07	13.2	2.60	17.7	3.77
Q0.1%
$C_v=0.1$	2.2	0.37	4.59	0.91	6.16	1.32
$C_v=0.6$	4.97	0.84	10.37	2.05	13.94	2.98
$C_v=1$	5.53	0.94	11.54	2.28	15.51	3.31
$C_v=2$	6.04	1.03	12.6	2.49	16.93	3.62
$C_v=4$	6.33	1.08	13.21	2.61	17.75	3.79
Q1%
$C_v=0.1$	1.63	0.28	3.41	0.68	4.59	0.99
$C_v=0.6$	4.46	0.76	9.34	1.86	12.56	2.71
$C_v=1$	5.18	0.88	10.84	2.16	14.58	3.15
$C_v=2$	5.9	1.0	12.33	2.46	16.59	3.58
$C_v=4$	6.33	1.08	13.24	2.64	17.81	3.84
Q5%
$C_v=0.1$	1.13	0.19	2.37	0.48	3.2	0.7
$C_v=0.6$	3.79	0.65	7.97	1.61	10.73	2.36
$C_v=1$	4.67	0.80	9.82	1.98	13.23	2.91
$C_v=2$	5.66	0.97	11.89	2.4	16.03	3.52
$C_v=4$	6.33	1.08	13.3	2.68	17.92	3.93

shows these systematic biases in estimating the parameters of the Gumbel distribution.

2.2.2. The Method of Linear Moments

In the case of L-moments, the systematic biases are influenced by the coefficient of L-variation ${\tau }_2$, which is the counterpart of the MOM-specific coefficient of variation but estimated using the first two linear moments. The coefficient of L-variation always takes values between 0 and 1 [70,93]. In this situation, four values (0.1, 0.4, 0.6, and 0.8) were chosen to include a diversified variability from small to large. Figure 3 shows the results of the inverse functions obtained for the analyzed cases.

As in the case of the method of ordinary moments, the theoretical biases will be highlighted both in the estimation of the parameters and in the estimation of the quantile values related to the event with the average return period T = 10,000.

Table 4. The theoretical biases for the scale parameter: L-moments.

The Scale Parameter $\mathit{\beta }$
The Coefficient of L-Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Theoretical Bias [%]
${\tau }_2=0.1$	3.6	−0.52	8.04	−1.57	11.03	−2.51
${\tau }_2=0.4$	3.6	−0.52	8.04	−1.57	11.03	−2.51
${\tau }_2=0.6$	3.6	−0.52	8.04	−1.57	11.03	−2.51
${\tau }_2=0.8$	3.6	−0.52	8.04	−1.57	11.03	−2.51

and

Table 5. The theoretical biases for the position parameter: L-moments.

The Scale Parameter $\mathit{\gamma }$
The Coefficient of L-Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Theoretical Bias [%]
${\tau }_2=0.1$	−0.01	0.11	0	0.32	0.01	0.52
${\tau }_2=0.4$	−0.04	0.58	−0.01	1.76	0.06	2.84
${\tau }_2=0.6$	−0.08	1.16	−0.01	3.52	0.12	5.69
${\tau }_2=0.8$	−0.16	2.31	−0.02	7.03	0.25	11.37

show the biases related to the two parameters of the distribution.

Table 6. The theoretical biases for rare events (p = 0.01%, 0.1%, 1%, and 5%): L-moments.

Q0.01%
The Coefficient of L-Variation	Number of Values
	$\mathit{n}=80$		$\mathit{n}=25$		$\mathit{n}=15$
	Empirical Probability
	Weibull	Hazen	Weibull	Hazen	Weibull	Hazen
	Systematic Bias [%]
${\tau }_2=0.1$	2.09	−0.27	4.72	−0.8	6.5	−1.29
${\tau }_2=0.4$	3.19	−0.4	7.14	−1.19	9.81	−1.91
${\tau }_2=0.6$	3.39	−0.42	7.57	−1.26	10.39	−2.02
${\tau }_2=0.8$	3.49	−0.44	7.8	−1.3	10.7	−2.08
Q0.1%
${\tau }_2=0.1$	1.87	−0.22	4.19	−0.66	5.75	−1.06
${\tau }_2=0.4$	3.08	−0.36	6.89	−1.09	9.46	−1.74
${\tau }_2=0.6$	3.32	−0.39	7.42	−1.17	10.19	−1.87
${\tau }_2=0.8$	3.45	−0.41	7.72	−1.22	10.6	−1.95
Q1%
${\tau }_2=0.1$	1.51	−0.16	3.38	−0.47	4.64	−0.75
${\tau }_2=0.4$	2.87	−0.3	6.43	−0.9	8.83	−1.43
${\tau }_2=0.6$	3.19	−0.33	7.14	−1.0	9.18	−1.59
${\tau }_2=0.8$	3.38	−0.35	7.56	−1.06	10.39	−1.68
Q5%
${\tau }_2=0.1$	1.14	−0.09	2.56	−0.28	3.52	−0.45
${\tau }_2=0.4$	2.58	−0.21	5.79	−0.63	7.96	−1.01
${\tau }_2=0.6$	3.0	−0.25	6.73	−0.74	9.25	−1.17
${\tau }_2=0.8$	3.27	−0.27	7.33	−0.8	10.07	−0.127

presents the systematic biases obtained with L-moments for the value of the quantile related to the annual exceedance probabilities of interest.

2.3. Choosing the Best Model

In general, choosing the best model involves two components, namely one subjective and one objective. For the vast majority of parameter estimation methods (MOM, MLE, PWM, LSM, WLSM, ENT, MIX), the subjective component is the predominant one because there are no rigorous criteria for choosing the best model, it being chosen based on the results of the application of indicators and performance tests, the results of which are applicable and can only be interpreted in the area of the annual probabilities exceeding the observed values. Outside of this field (in general, data are wanted to be forecast there), they can no longer constitute a selection criterion, because they are based only on the difference between generated and observed (recorded, real) values.

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 (${\tau }_3$) and L-kurtosis (${\tau }_4$) of the observed data [4,67,68,69,71,73,74,75,89,90,91,92,93,94,95,96]. Unfortunately, the Gumbel distribution is not defined, like the two-parameter Log-normal or Gamma distribution, by a variation curve of these indicators, but it has constant values regardless of the observed data analyzed [93]. On the general graphs of variation of the indicators obtained with the L-moments method, this is defined by a point [4,67,93].

Thus, in the case of FFA, the selection of the best model is based on the selection criteria of the L-moments method, while in the case studies regarding the maximum determination of precipitation, the selection is based on both the L-moments criteria and performance indicators, because the values of the quantiles related to the interested probabilities are approximated by the empirical ones of the recorded data; in general, there are recorded data in this field.

3. Case Studies and Data

3.1. Flood Frequency Analysis

The frequency analysis of the maximum flows consists of determining the annual maximum values for three case studies.

The data related to the rivers Bahna, Nicolina, and Siret represent annual maximum values that characterize each year of analysis (Annual Maximum Series).

The Siret and Nicolina rivers are located in eastern and southeastern Romania, while the Bahna River, a left tributary of the Danube river, is located in the southwestern part of Romania (see Figure 4).

Peculiarities specific to the analyzed sites were highlighted in previous materials [93].

The graphs of the chronological maximum annual values are presented in Figure 5.

The boxplot representations of the analyzed series are presented in Figure 6. The values of the 25%, 50%, and 75% quartiles are highlighted, as well as the minimum and maximum values.

The statistical characteristics of the analyzed data, including the expected value ($\mu$), the coefficient of variation ($C_v$), the skewness coefficient ($C_s$), the kurtosis coefficient ($C_k$), the first four linear moments ($L_1$, $L_2$, $L_3$, $L_4$), the coefficient of L-variation (${\tau }_2=L_2/L_1$), the L-skewness (${\tau }_3=L_3/L_2$), and the L-kurtosis (${\tau }_4=L_4/L_2$), are presented in

Table 7. Information regarding the statistical indicators of the series: MOM and L-moments.

River	Number of Records “n”	Statistical Indicators
	[yr]	$\mathit{\mu }$	$\mathit{\sigma }$	${\mathit{C}}_{\mathit{v}}$	${\mathit{C}}_{\mathit{s}}$	${\mathit{C}}_{\mathit{k}}$	${\mathit{L}}_1$	${\mathit{L}}_2$	${\mathit{L}}_3$	${\mathit{L}}_4$	${\mathit{\tau }}_2$	${\mathit{\tau }}_3$	${\mathit{\tau }}_4$
		[m3/s]	[m3/s]	[-]	[-]	[-]	[m3/s]	[m3/s]	[m3/s]	[m3/s]	[-]	[-]	[-]
Bahna	30 (1992–2020)	13.3	20.2	1.519	3.108	10.04	13.3	8.07	4.91	3.52	0.608	0.608	0.436
Nicolina	39 (1979–2017)	14.1	16.8	1.193	2.796	9.44	14.1	7.55	3.60	2.22	0.536	0.477	0.294
Siret	39 (1970–2008)	1443	915	0.634	1.413	5.87	1443	490	112	90.6	0.339	0.228	0.185

.

For the LH-moments method, the statistical indicators’ values are presented in

Table 8. Information regarding the statistical indicators of the series: LH-moments.

Station	Statistical Indicators
	${\mathit{L}}_{\mathit{H}1}$	${\mathit{L}}_{\mathit{H}2}$	${\mathit{L}}_{\mathit{H}3}$	${\mathit{L}}_{\mathit{H}4}$	${\mathit{\tau }}_{\mathit{H}2}$	${\mathit{\tau }}_{\mathit{H}3}$	${\mathit{\tau }}_{\mathit{H}4}$
	[mm]	[mm]	[mm]	[mm]	[-]	[-]	[-]
LH-moments—level 1
Bahna	21.3	9.73	5.62	3.68	0.456	0.577	0.378
Nicolina	21.6	8.36	3.88	2.34	0.386	0.464	0.28
Siret	1932	451	135	89.9	0.233	0.299	0.199
LH-moments—level 2
Bahna	27.8	11.2	6.11	3.85	0.401	0.548	0.345
Nicolina	27.2	9.0	4.11	2.44	0.331	0.456	0.27
Siret	2233	442	148	90.7	0.198	0.334	0.205

.

3.2. Annual Maximum Daily Rainfall

The data recorded for the two stations (Dângeni and N. Balcescu) are the annual daily maximums (AMS).

The stations are located in the northeast (Dângeni) and the east (N. Balcescu) of Romania (see Figure 7).

Figure 8 shows the chronological series of maximum annual values over the entire analysis period.

For the Dângeni Station, the maximum value recorded annually (per 24 h) is approximately 110 mm, while for the N. Balcescu Station the maximum value recorded is 90 mm.

Statistical information regarding the 25%, 50%, and 75% quartiles and the minimum and maximum values are highlighted graphically in Figure 9.

The statistical indicators of the analyzed series, which are necessary to estimate the parameters using MOM and L-moments, are presented in

Table 9. The statistical indicator values of the series: MOM and L-moments.

Station	Number of Records “n”	Statistical Indicators
	[yr]	$\mathit{\mu }$	$\mathit{\sigma }$	${\mathit{C}}_{\mathit{v}}$	${\mathit{C}}_{\mathit{s}}$	${\mathit{C}}_{\mathit{k}}$	${\mathit{L}}_1$	${\mathit{L}}_2$	${\mathit{L}}_3$	${\mathit{L}}_4$	${\mathit{\tau }}_2$	${\mathit{\tau }}_3$	${\mathit{\tau }}_4$
		[mm]	[mm]	[-]	[-]	[-]	[mm]	[mm]	[mm]	[mm]	[-]	[-]	[-]
Dângeni	49 (1969–2017)	47.1	19.9	0.424	1.033	1.103	47.1	10.9	2.19	1.67	0.233	0.200	0.153
N. Balcescu	56 (1962–2017)	48.5	17.54	0.361	0.380	−0.36	48.5	10.0	0.69	0.74	0.206	0.069	0.074

.

In the case of the LH-moments method, the values of the statistical indicators necessary to estimate the parameters of the distributions are presented in

Table 10. The statistical indicator values of the series: LH-moments.

Station	Statistical Indicators
	${\mathit{L}}_{\mathit{H}1}$	${\mathit{L}}_{\mathit{H}2}$	${\mathit{L}}_{\mathit{H}3}$	${\mathit{L}}_{\mathit{H}4}$	${\mathit{\tau }}_{\mathit{H}2}$	${\mathit{\tau }}_{\mathit{H}3}$	${\mathit{\tau }}_{\mathit{H}4}$
	[mm]	[mm]	[mm]	[mm]	[-]	[-]	[-]
LH-moments—level 1
Dângeni	58.1	9.87	2.58	1.38	0.17	0.261	0.14
N. Balcescu	58.5	8.02	0.95	0.84	0.137	0.119	0.104
LH-moments—level 2
Dângeni	64.6	9.4	2.63	1.28	0.146	0.279	0.136
N. Balcescu	63.9	7.0	1.16	1.0	0.109	0.166	0.143

.

4. Results and Discussions

The results and the discussions regarding the obtained results are mainly focused on two important directions. (1) Verification of the applicability of the Gumbel distribution in the field of maximum flow frequency analysis (FFA), where the analysis focuses on the quantile values in the field of low exceedance probabilities (p < 1%), because in FFA these are the values that want to be forecasted (“impossible floods”). Both in Romanian and international legislation (ICOLD) regarding the design of hydrotechnical constructions, it is necessary to directly determine the values of the quantiles of the following annual exceeding probabilities: 0.01%, 0.1%, 0.5%, 1%, 2%, 5%, 10%. This depends on the importance class of the construction [99]. (2) Verifying the applicability of the Gumbel distribution in the direct determination of the maximum precipitation values for return periods of up to 500 years (p = 0.2%), values generally required for the construction of IDF curves.

4.1. Flood Frequency Analysis

In all of the case studies, the values generated by the Gumbel distribution are analyzed compared to the Burr reference distribution and the L-moments reference method, for the reasons stated in the previous sections. The Burr distribution has a large number of parameters, managing to properly calibrate all of the linear moments, and the L-moments method is superior to the other estimation methods. Additionally, for the rigor of the analysis, another distribution with a high number of parameters is used, namely the Wakeby distribution.

4.1.1. Verification of Normality

The verification of the normality of the data was performed graphically (see Figure 10) to be able to easily notice that the observed data do not come from a normal distribution. The red line represents the normality of the observed data highlighted by the blue dot.

In all three cases, it can be observed that the recorded values have a heavy-tailed tendency.

4.1.2. The Verification of Stationarity and Outliers

Stationarity was checked using the “t” test. Its results as well as the critical values are presented in

Table 11. The results of the stationarity check.

Series	t-Test
	Results	Critical Value (10%)
Bahna	1.405	2.048
Nicolina	0.252	2.026
Siret	1.708	2.026

. No non-stationarity of the analyzed data is observed.

Outliers were checked using the Grubbs test; no outliers were identified in the analyzed data series.

4.1.3. Statistical Analysis of the Field of Recorded Data

The graphic verifications of the correlation (Q-Q plot) of the observed data with those generated by the analyzed distributions with the estimated parameters using the L-moments method are presented in Figure 11, Figure 12 and Figure 13. Both the X-axis and the Y-axis represent the observed data values.

It can be observed that in all three case studies the Burr, GEV, and Wakeby distributions have the best results, with the values generated by them being the closest to those of the recorded data. The additional use of some tests (Kolmogorov–Smirnov, Anderson Darling, etc.) and performance indicators (RAE, RME, RMSE, etc.) would highlight the same situation observed graphically.

4.1.4. Statistical Analysis of the Field of Low Annual Exceedance Probabilities

Considering the values of the quantiles related to the annual exceedance probabilities that are to be determined, this analysis is the most important. It also represents the field in which there is diverse and different behavior of statistical distributions, imposed by the type of distribution (number of parameters and the family it belongs to), the parameter estimation method, and the available lengths of the observed data.

In the case of the three case studies, the variation graphs of the inverse function (quantile function) related to each distribution and the estimation methods of the analyzed parameters are presented in Figure 14.

It can be seen that the values generated by the Gumbel distribution, regardless of the parameter estimation method used, are much lower than the GEV, Burr, and Wakeby distributions, which have a larger number of parameters, thus managing to calibrate the higher-order linear moments.

Analyzing the values generated for the annual probability of exceeding 0.1%, it can be seen that the quantile values of the Gumbel distribution vary in the range of 5000–6500 mc/s for the Siret River, between 38 and 130 m³/s for the Bahna River, and between 40 and 120 m³/s in the case of the Nicolina River. In all analyzed cases, the lower values were generated using the entropy method (ENT), while the higher values were generated by estimating the parameters with the second-order linear moments method. For the same annual probability of non-exceedance (0.1%), the values generated through the distribution GEV, having parameters estimated with the L-moments method, are 7157 m³/s for the Siret River, 429 m³/s for the Bahna River, and 263 m³/s for the Nicolina River. The Burr (four-parameter) and Wakeby (five-parameter) distributions generated relatively close values due to the possibility of these distributions to calibrate all of the linear moments specific to the L-moments method. In the case of the Siret River, the generated values are between 7498 m³/s (Burr) and 8026 m³/s (Wakeby). In the case of the Bahna River, the values are between 379 m³/s (Wakeby) and 400 m³/s (Burr). For the data series related to the Nicolina River, the Burr distribution generated a value of 239 m³/s, and the Wakeby distribution generated a value of 220 m³/s.

It can be observed that the use of the Gumbel distribution, without respecting the calibration criteria imposed by the L-moments method, generates values characterized by very large errors for the values of the quantiles related to some rare events that want to be forecasted, especially if we take into account that generally, in FFA, the direct determination of some events with a return period of up to 10,000 years is required, which leads to even greater forecast errors. Thus,

Table 12. Relative errors of the Gumbel distribution compared to the Burr distribution.

River	Relative Errors [%]
	Q0.01%	Q0.1%	Q1%	Q2%	Q5%
Siret	−47.6	−26.8	−10.6	−6.6	−2.0
Bahna	−1217	−359.5	−77.1	−37.0	−0.2
Nicolina	−498.6	−187.4	−51.8	−28.8	−6.2

shows the estimation errors of the quantile of the Gumbel distribution, having as a reference the values generated by the Burr distribution (considered in these cases the “parent” distribution). For these case studies, the theoretical biases (relative errors from the behavior of the Gumbel distribution depending on the length of the available data series) are not highlighted because they are insignificantly smaller if we compare them to the errors from the selection of the best model. Very large errors can be observed, which, in the case of inadequate analyses and in the absence of rigor in these analyses, can lead to the defective dimensioning of some hydrotechnical works, which can lead to undesirable consequences, including economic losses and, most importantly, losses of human lives. This reasoning is also valid in the case of three-parameter distributions (Pearson III, GEV, Pareto, etc.) when they are not used and applied properly.

4.2. Annual Maximum Daily Rainfall (24 h)

In the case of the analysis of the maximum annual precipitation, the values of the interested quantiles (h_p) are those related to a maximum annual exceedance probability of 0.2% (T = 500 years), 0.5% (T = 200 years), 1% (T = 100 years), and 2% (T = 50 years).

In general, for this interval of probabilities, the data series are long enough that the errors in estimating the values with the Gumbel distribution are small. But, there are also cases where the lines are not long enough, requiring a more laborious analysis in choosing the best model.

Thus, in this section, the criteria for choosing the best distribution are in compliance with the conditions imposed by the L-moments method (also chosen in these cases as a reference) but also in the use of some performance indicators that are based on highlighting the relative errors between recorded and forecasted values. For the two analyzed case studies, the RME (Relative Mean Error) and RAE (Relative Absolute Error) performance indicators are used.

4.2.1. Verification of Normality

The normality of the data is verified graphically and presented in Figure 15. It can be observed that the data do not come from a normal distribution.

In both cases, the observed values have a heavy-tailed tendency.

4.2.2. The Verification of Stationarity and Outliers

Regarding the stationarity check, the results of the “t” test are presented in

Table 13. The results of the stationarity check: Dângeni and N. Balcescu Stations.

Series	t-Test
	Results	Critical Value (10%)
Dângeni Station	1.995	2.012
N. Balcescu Station	0869	2.005

, with the values being lower than the critical ones, thus highlighting the stationarity of the analyzed data. An analysis was also carried out regarding the existence of outlier values (Grubb’s test), with no such values being identified.

4.2.3. Analysis of Forecasted Values

Figure 16 presents the results and behaviors of the inverse functions of the analyzed distributions.

Upon analyzing the obtained results, it can be easily observed that the values generated by the four distributions differ significantly for both case studies. The particular aspects of the obtained results are detailed in the next section.

Dângeni Station Results

In the case of the Dângeni Station, the values generated with the Gumbel distribution (reference probability p = 0.2%) are around the value of 136 mm, except the value related to the estimate with the ENT method, where the predicted value is 160 mm, but it can be easily observed that it practically does not pass through the points of the recorded values.

The values generated by the distributions with a high number of parameters vary between 147 mm (GEV distribution) and 162 mm (Burr and Wakeby distributions).

The results of the performance indicators are presented in the

Table 14. Distributions’ performance values: Dângeni Station.

Distribution	Parameter Estimation Method	Statistical Measures
		RME	RAE
Gumbel	MOM	0.0061	0.0311
	L-mom	0.0065	0.0323
	LH1-mom	0.0081	0.0366
	LH2-mom	0.0092	0.0417
	PWM	0.0061	0.031
	MLE	0.006	0.0321
	LSM	0.0072	0.0337
	WLSM	0.0069	0.0331
	ENT	0.0167	0.0935
	MIX	0.0145	0.0823
GEV	L-mom	0.0056	0.0294
Burr	L-mom	0.0069	0.0373
Wakeby	L-mom	0.0075	0.0399

. Upon analyzing the values, the GEV distribution has the best result. But, considering that this is a relevant indicator only in the area of the probabilities of the recorded values, and given that the empirical probability related to the highest value (n = 49) is 1.02%, the selection of the best model must be made while respecting the criteria imposed by the reference method, L—moments. Thus, following the analysis of the L-skewness and L-kurtosis statistical indicators, the Burr and Wakeby distributions are the ones that properly calibrate the similar values of the observed data, namely, 0.2 and 0.153, respectively.

The relative errors between the values generated by the Gumbel distribution and those of the best model are presented in

Table 15. Relative errors of the Gumbel distribution compared to the Burr distribution: Dângeni Station.

Station	Relative Errors
	h0.2%	h0.5%	h1%	h2%
Dângeni	−18.9%	−10%	−5%	−1.3%

. They vary between −18.9% and −1.3% depending on the predicted probability.

N. Balcescu Station

Like the Dângeni Station, the values of interest are those of the rarest event, namely the one with an annual probability exceeding 0.2%. The Gumbel distribution generated values between 128 mm (LH-moments and MIX method) and 164 mm (ENT method). Even in this case, it can be observed that the values generated with the ENT method do not properly approximate the recorded values. For all parameter estimation methods, the values generated by the Gumbel distribution are superior to those generated by the GEV, Burr, and Wakeby distributions. In the case of the GEV distribution, the corresponding p = 0.2% value is 105 mm, while in the values generated by the Burr and Wakeby distributions, it is around 120 mm. Graphically, it can be seen that the curves of the three distributions pass through the points related to the observed data.

The performances of the distributions are presented in

Table 16. Distributions’ performance values: N. Balcescu Station.

Distribution	Parameter Estimation Method	Statistical Measures
		RME	RAE
Gumbel	MOM	0.0102	0.0575
	L-mom	0.0089	0.0512
	LH1-mom	0.0131	0.0716
	LH2-mom	0.0196	0.104
	PWM	0.0093	0.0528
	MLE	0.0088	0.051
	LSM	0.0133	0.0585
	WLSM	0.0096	0.0533
	ENT	0.0186	0.0947
	MIX	0.0227	0.1608
GEV	L-mom	0.0091	0.0477
Burr	L-mom	0.0060	0.0340
Wakeby	L-mom	0.0061	0.0346

. Based on the results, the best model is the Burr distribution. This choice is also in accordance with the corresponding calibration of the higher-order statistical indicators (L-skewness and L-kurtosis) specific to the L-moments method, namely 0.069 and 0.074, respectively. Very close values of RAE and RME are also generated by the Wakeby distribution, an aspect otherwise expected as both distributions fulfill specific calibration criteria of the L-moments method, an aspect partially due to the fact that the empirical probability related to the maximum value of the observed data (n = 56) is 0.893%, a value closer to the desired 0.2%, the data extrapolation interval being smaller.

Table 17. Relative errors of the Gumbel distribution compared to the Burr distribution: N. Balcescu Station.

Station	Relative Errors of the Gumbel Distribution Compared to the Burr Distribution
	h0.2%	h0.5%	h1%	h2%
N. Balcescu	15.3%	14.3%	7.9%	10.5%

shows the errors between the values generated by the Gumbel distribution and those generated by the best model, namely the Burr distribution. It can be seen that the errors in the estimation of the best model are between 10.5% and 15.3%, increasing with the decrease of the annual probability of exceeding.

5. Conclusions

The Gumbel distribution was, is, and will probably remain one of the most commonly used statistical distributions in the analysis of extreme events in hydrology.

In the literature, it is used using different parameter estimation methods, among which the most common are the method of ordinary moments and the method of linear moments. Its applicability on a large scale is generally due to the simplicity of the equations needed to estimate the parameters, as well as the simplicity of the expression of the inverse function, being generally applied using the characteristic frequency factor.

Following the case studies presented in this manuscript, which include three frequency analyses for determining the maximum flows and two frequency analyses of the maximum annual precipitation, with the parameters estimated with nine methods, as well as following the observations based on the available scientific materials, it can be concluded that the real utility of the distribution is limited, and its application can only be made if the conditions imposed by the parameter estimation methods are met.

Following the theoretical analysis and the presented case studies, it was noted that the Gumbel distribution is strongly influenced by the variability of the available data lengths, the forecasted values being characterized by large uncertainties. These aspects are highlighted in the comparative analysis with results obtained with more rigorous distributions, such as the Burr and Wakeby distributions.

Taking into account that among all of the parameter estimation methods, the L-moments method proved to be the most robust and reliable method, being also the only method in the regionalization analyses of extreme events in hydrology, it is recommended to use the Gumbel distribution with this method and only after a preliminary analysis regarding the most accurate calibration of the statistical indicators, L-skewness and L-kurtosis, as characteristics of the distribution with those of the analyzed data series. Compared to other two-parameter distributions [93], the Gumbel distribution has the great disadvantage that the values of these indicators do not fit on a variation curve of interdependence but have constant values (L-skewness ${\tau }_3=0.17$ and L-kurtosis ${\tau }_4=0.15$).

The limited flexibility and adaptability necessitate careful use of the Gumbel distribution in hydrology, climate science, and other fields dealing with the analysis of extreme values. Careful analysis of data quality, statistical characteristics of observed data, and parameter estimation methods is mandatory for effective application of this distribution.