Extreme Value Distributions: An Overview of Estimation and Simulation

The generalized extreme value distribution (GEVD) and various extreme value distributions are commonly applied in air pollution, telecommunications, operational risk management, finance, insurance, material sciences, economics, and hydrology, among many other industries that deal with extreme events. Extreme value distributions (EVDs) typically limit the distribution of maximum and minimum values for many random observations drawn from the same arbitrary distribution. Besides that, it is a crucial method for forecasting future events and emerged as critical method for predicting future events. As a result, prior research is required to select the best estimation method to obtain a reliable value for the parameters of extreme value distributions. This study provides an overview of three-parameter estimation methods based on goodness-of-fit statistics and root mean square error (RMSE). This paper reviewed and compared three estimation methods used to approximate values of parameters for simulated observations taken from the EVD and GEVD. The method of moments (MOMs), maximum likelihood estimator (MLE), and maximum product of spacing (MPS) were the methods investigated in this study. Our findings indicated that the MPS performed better based on the mean square errors (MSEs); meanwhile, the MPS had similar goodness-of-fit statistic values compared to the MLE.


Introduction
Extreme value distribution (EVD) is used to limit distributions for maximum or minimum [1]. us, as the sample size increases with the smallest or largest data in independent identically distributed random variables, the data set density shape will follow one of the three types of EVD [1,2]. EVD is also used to model tail-related risk measurements such as value at risk, return level, or expected shortfall [3]. Extreme wind speed analysis is used mainly in natural emergency preparedness, mitigation, management, prevention, and various civil engineering, environmental, and ocean applications [4]. An accurate estimate of the parameters for any analyses using the EVD is a must. Hence, there should be a suitable estimation method that provides accurate estimates for the parameters of the EVD. ere had been many studies related to various parameter estimation methods on EVD.
Without a doubt, parameter estimation is essential to t any probability distribution on any data sets. As a result, various estimation methods could provide us with insight into determining the "best-tting" distribution and estimate the parameters for EVD, such as the scale, shape, and location parameters. e following are some standard parameter estimation methods that are commonly used in probability distribution tting: (i) e MOM (Johan Bernoulli, 1667-1748).
Since their introduction, these methods have progressed through several stages and have their drawbacks and bene ts [5][6][7]. Nonetheless, the MLE method is the most widely used estimation method. e three methods mentioned above are used in this study to estimate EVD and GEVD parameters. Several studies were comparing the various estimation methods for different distributions. By reviewing other studies, we described the basic idea of each estimation method and their applications on EVD. A simulation study was carried out for reference purposes to assess the performance of the estimators. As it has been widely deployed in many research areas, the EVD is used to represent the distributions of various observations. ese include wind speed and energy data [4,[8][9][10][11][12], wave data prediction [13], data on air pollution [14][15][16][17][18], information and communication technology [19], data on flooding [20], financial risk [3,21], temperature [22], food drying technology [23], and rainfall [24]. It has also been implemented in public health and medical sciences [25,26]. erefore, studies comparing MLE, MOM, and MPS estimators for GEVD, two-parameter EVD, and three-parameter EVD were reviewed in this research. e MOM method is the oldest method for estimating parameters, whereas the MLE is the most commonly used. However, MLE can fail in various circumstances, necessitating a less popular alternative (i.e., MPS). is review article aimed to guide selecting the best estimation method for the GEVD and EVD, which will be of great interest to applied statisticians. e novelty of this review stems from the fact that no thorough review of MOM, MLE, and MPS estimators for EVD has been made. e following is how this article was structured. e history of EVD is presented before reviewing the MLE, MOM, and MPS. Next, EVD applications were discussed, followed by a simulation study. Last but not least, conclusions were drawn based on the factors reviewed and discussed above.

Extreme Value Theory (EVT) and Extreme Value Distribution (EVD)
An extreme value in a series of observations is either a very large or small value. It can even be described as the outer or outlier points, which are the highest and lowest values. EVT is a theory of modeling and measuring events with the least amount of probability [27]. To be specific, EVT identifies extreme events based on a probability of occurrence and also depicts the extreme events through statistical analysis of the extreme properties. It consists of 3 types of distributions. It only requires three distributions to model the maximum or minimum random observations for the same distribution [2]. Recently, the EVT has emerged as one of the most important statistical disciplines for engineers and applied scientists [14,28]. If we assume X 1 ; X 2 ; . . ., X n are independent random variables with a standard distribution function (F). en, M n � Max X i ; . . . ; X n for each i with i � 1, . . . , n denotes the maximum of observational process over n time units of observations. According to Coles [28], the distribution of M n can be derived as follows: (2) e probability density function (PDF), f X (x), for EVD distribution derived from the cumulative function F X (x) can be derived as below: ere is a concern with degeneration of the exact function because the distribution function, F, is unknown, and n ⟶ ∞. Hence, we pursue approximate families of models for F n that can be estimated solely on the extreme data. As per the central limit theorem (CLM), the estimation is similar to the usual practice of approximating the sample means for normal distribution. To resolve this situation, we developed a normalized version of M n to stabilize the function. A normalized M n could be generated as below with the presence of normalizing constants, a n and b n : e relevant and suitable choices of a n and b n stabilize the location and scale of M n as n ⟶ ∞. M * n converges in the form of three EVD distribution types: Type I, Type II, and Type III. If the normalizing constants a n and b n exist, thus, G(x) is the nondegenerate cumulative distribution function (CDF), which relates to the three EVD families: Type I, Type II, and Type III. (Type I). Emil Gumbel, a German mathematician, invented the Gumbel distribution . e primary focus was on the extensive use of the EVT in various fields for modeling extreme events [2]. e formula includes the following PDF:

Gumbel Distribution
whereby σ � distribution scale (σ > 0) and μ � location parameter. e CDF can then be given as follows: (Type II). A French mathematician, Maurice Fréchet (1878-1973), had derived the Fréchet Distribution. In 1927, he proposed one possible limiting distribution for the maximal order statistics [2]. e Fréchet distribution is also known as the inverse Weibull distribution (IWD). It includes the following CDF and PDF:

Weibull Distribution
e CDF and PDF for this distribution are as below: 2.3.1. Two-Parameter Weibull Distribution. PDF is as follows: CDF is as follows: 2.3.2. ree-Parameter Weibull Distribution. PDF is as follows: CDF is as follows: whereby μ � location parameter (μ � 0 for the two-parameter Weibull Distribution), σ � scale parameter (σ > 0), and α � shape parameter (α > 0). e three EVD families can be generalized to form a single distribution called the generalized extreme value distribution (GEVD). e GEVD was an extension of the EVT developed by Fisher-Tippett (1928) and Gnedenko (1943). It is a good choice for representing the distribution of the minimum and maximum sequences of independent identically distributed random variables [1,2].

GEVD.
e CDF for the three-parameter is as follows: From equation (17), σ and 1 + α(x − μ)/σ > 0, where μ and α can take any real value. e three types of EVD can be obtained through GEVD based on the value of alpha where α � 0 is the Type I EVD (Gumbel distribution), α > 0 is the Type II EVD (Fréchet distribution), and α < 0 is the Type II EVD (Weibull distribution).
Meanwhile, for the PDF for GEVD is given in equation (18) with σ > 0, with α and μ, can take any real value.

Application Study Review
In this section, we will review and discuss the comparison of MOM, MLE, and MPS estimation methods using actual data or simulation studies as the following: Hall et al. [34] estimated the generalized Gumbel distribution parameters using the MLE method in 1989.
A comparison study was conducted between the standard MLE and the unbiased MLE estimator, which is derived from MLE linear functions, product spacing method, and quantile estimate method to estimate two exponential distribution parameters. For both the location and scale parameters, the unbiased MLE had the lowest RMSE, followed by MPS and MLE. Overall, both methods performed nearly identically equivalent. However, the unbiased MLE provides better parameter estimates [35].
Hurairah et al. [36] proposed a new Gumbel distribution for handling air pollution data by introducing a new parameter that shapes the parameter α. e MLE method is applied to estimate the parameters of the new Gumbel distribution. e simulated results indicated that the new Gumbel distribution could achieve higher accuracy in fitting carbon monoxide (CO) data and significantly impacting air pollution studies [14].
Other research also used the MLE to estimate the following parameters: Gumbel, generalized Pareto distribution with two and three parameters, Weibull with two and three parameters, and GEVD [17]. e two-year daily maximum data were used to analyze the efficiency of the six distributions using error and accuracy measures as performance indicators.
e GEVD was found to be an adequate distribution for maximum daily density of particulate matter (PM10) for all monitoring stations under study. MOM was used in another study to estimate the parameters of the Gumbel and Fréchet distribution instead of lognormal to fit the daily maximum concentration of PM10 in Malaysia. e goodness-of-fit was used to select the distribution that best fits the data for PM10 exceedances based on the Malaysian Ambient Air Quality Guidelines (MAAQG). e work concluded that the EVD fits the actual high value of PM10 better than central fitting distribution [16].
On the other hand, Wong and Li [37] compared the MLE and the MPS in estimating parameters of EVD using samples with small sample sizes. His study found that the MPS functioned satisfactorily. Not only does it performs consistently for data maxima extracted from clusters, but it also accurately estimated more data generated from a known parameter set, whereas the MLE does not. Based on this finding, the MPS is considered one of the best estimation methods for fitting EVD.
Jiang [38] had demonstrated that the location and scale estimator parameters were biased, and MPS underestimated the shape parameter. Hence, he modified an MPS to fit a three-parameter Weibull distribution that could accurately estimate parameters better. Meanwhile, Huang and Lin [39] also altered the MPS method to improve the estimate parameters of the GEVD. e simulations revealed that not only is the suggested method highly efficient and applicable across the entire parameters, but it also outperforms the study's existing parametric and nonparametric methods.
A least square estimation (LSE), MLE, and MPS were used to compare traditional estimation methods to fit the generalized inverted exponential distribution [40]. e study was also intended to analyze the estimates' behavior for small samples. Results showed that MPS outperformed the other two methods with a minor mean square error (MSE). erefore, the study suggested using MPS since it exceeded both MLE and LSE.
Akram and Hayat [41] compared the performance of fitting a three-parameter Weibull distribution with the following parameter estimation methods in terms of bias and RMSE in a small sample: L-moments, LSE, the modified MLE, MOM, and MPS. Overall, the L-moments method performed well and is the best estimation method. e modified MPS performed well when the shape parameter was less than a specific value. In contrast, the modified MLE method was inefficient and inconsistent because it might not exist.
Next, Soukissian and Tsalis [4] investigated parameter estimation methods for predicting extreme wind speeds in the Atlantic and Pacific ocean basins. A natural wind measurements and simulation study from four buoys were used in the analysis. According to the research, the MPS, elemental percentile (EP), and standard entropy method appeared less accurate than the MLE. Based on the MSE, bias, and variance of the estimated data, the MLE was a much better estimation method.
Meanwhile, Salah et al. [42] used various estimation methods such as weighted least squares, MLE, probabilityweighted moments, and LSE for the accelerated life test (ALT) under the family of exponentiated distributions. He chose the best method to estimate the reliability function. e four methods were applied using both simulated and actual-world data. Among other estimation methods, it has been discovered that the MLE produces the best results. Louzada et al. [43] considered the MLE, modified moments, MOM, L-moments, minimum distance estimator percentile estimation, MPS, ordinary, and weighted least squares for estimating unknown parameters of the extended exponential geometric distribution. Compared to its competitors, the MPS estimated the best for the extended exponential geometric distribution parameters.
Singh et al. [44] studied the possibility of estimating the scale and shape parameters for the generalized inverted exponential distribution using progressive type-II censored samples. e MPS was used to estimate the reliability, hazard functions, and parameters of the model. Based on a Monte Carlo simulation study, the MPS was compared to the corresponding MLE. Based on MSE, it is discovered that the MPS method outperforms the MLE. As a result, regardless of sample size, the former method could estimate reliability, hazard function, and distribution parameters well.
Dey et al. [45] investigated various methods and properties for estimating unknown parameters for the following distributions" (i) Exponentiated Chen distribution.
(iii) Exponentiated Gumbel distribution. e right-tail Anderson-Darling, MLE, percentile estimation, MOM, least squares estimation, Cramér-von-Mises, MPS, and Anderson-Darling methods were used in this study. Extensive simulation studies were used to compare them using Monte Carlo simulations. e results revealed that the MPS is the best estimator for transmuted Rayleigh and exponentiated Chen distributions in terms of biases and RMSE. e MLE method, on the other hand, is the best for estimating the exponentiated Gumbel distribution parameters [7,46,47]. e finite sample properties of the Marshall-Olkin extended exponential distribution parameters were obtained by ten estimation methods using Monte Carlo simulations. ey were Anderson-Darling, weighted least squares, L-moments, maximum likelihood, right-tail Anderson-Darling, ordinary least squares, modified moments, MPS, percentile estimation, and Cramér-von-Mises. e performance of all the methods was compared using the absolute, bias, and maximum absolute difference between RMSE and the estimated and actual distribution functions. e simulation demonstrated that the MLE and L-moments perform admirably in large sample sizes. Nonetheless, both methods have lower accuracy with small sample sizes than the MPS and Anderson-Darling methods [48].
e MPS was employed in the linear regression model based on Student-t, normal, skewed Student-t, and MLE distributions. A study found that all of the estimates were consistent and, in some cases, outperformed the MLE method. Furthermore, the MPS estimator is likely to exceed MLE when the sample size is small [49].
Vivekanandan [22] conducted Hissar extreme value analysis of rainfall and temperature using a logged Pearson Type-3 probability distribution and two-parameter lognormal fitted to one-day maximum and minimum rainfall and annual temperature series. L-moments, MLE, and MOM estimation methods were used to determine the distribution parameters based on their applicability. e study's tests revealed that the MLE estimated better than other methods for allocating the minimum and maximum rainfall and temperature.
Meanwhile, Nassar et al. [50] proposed a new extension for Weibull distribution. Two shape parameters and one scale parameter were included in the proposed distribution. It also contains submodels such as logarithmic-altered Weibull distribution and exponential distribution and the logarithmic-transformed exponential and logarithmictransformed Weibull distributions.
e research concentrated on the unknown parameters as well as several new mathematical properties. Least squares, MLE, percentilebased, MPS, and weighted-least square estimators have all been used. Monte Carlo simulations were used to compare the proposed estimation methods for large and small samples. Based on the results, percentile-based was the best performing estimator with respect to MSE. e applications on two actual data sets showed that the MPS performed better than the least square estimator for data set I. Meanwhile, the least square method is a better estimator for data set II.
Dey et al. [45] applied various estimation methods on the Gompertz distribution in a medical application. Fourteen methods were used to estimate the model parameters. A simulation study was conducted to compare these methods, and it was discovered that modified moments and moment estimators outperform others. Nonetheless, MPS estimators can still perform reasonably well and produce good results.
Last but not least, Ramos et al. [51] investigated the estimation of Fréchet distribution parameters. MLE, percentile estimators, MOM, L-moments, MPS, and ordinary and weighted-least squares were compared in this study, focusing on MSE. In terms of RMSE, the results revealed that MPS outperformed the other estimators significantly.

Parameter Estimation Methods
e parameters of EVD have been estimated using a variety of methods. Nonetheless, we will only concentrate on MLE, MOM, and MPS for the distributions mentioned above to evaluate the performance of each estimation method.

Maximum Likelihood Estimator.
Maximum likelihood estimator (MLE) is one of the methods used for estimating model parameters [5]. e MLE principle is to use the model with the highest likelihood. It is a necessary tool for many statistical modeling techniques and becomes a favored method of parameter estimation in statistics [52]. ere are three advantages of MLE [28]: it has desirable mathematical and optimality properties, it could give a consistent approach to parameter estimation problems, and it is applicable in almost all popular statistical software packages. An example of using MLE to estimate parameters for a probability distribution with 3 parameters μ, σ, and α is as follows: (i) Step I. e likelihood function of the probability distribution L(μ, σ, α) is obtained and written as follows: (ii) Step II. Take the natural log of the likelihood and collect terms involving μ, σ, α.
Step III. e differentiation of L(μ, σ, α) and solve it with respect to μ, σ, and α: e formulas for estimating μ, σ, and α for various extreme value distributions using MLE are shown in Table 1.

Method of Moments.
Method of moments (MOMs) is one of the conventional estimation methods for fitting statistical distributions [51]. e MOM estimators are usually easy to use and almost always produce some estimate. Unfortunately, MOM frequently generates estimators that could be improved. is method relies on matching the distribution moment to the sample moment. It is built on the presumption that sample moments should provide reasonable estimates of the corresponding population moments [53]. Equations in Table 2 show that x and S represent the sample mean and standard deviation, respectively. We define the mean value by μ � (1/n) n i�1 (X i ). e j th sample moment is then computed as follows: Journal of Probability and Statistics Journal of Probability and Statistics and the population moment by μ j (θ 1 , . . . , θ n ) � E(X) j , forj � 1, . . . , n, where θ 1 , . . . , θ n are unknown parameters. Next, m j � μ j (θ 1 , . . . , θ n ) is set and solved for θ 1 , . . . , θ n . e equations are the MOM's estimator for θ 1 , . . . , θ n . e formulas to estimate the parameters μ, σ, and α for various extreme value distributions using MOM are shown in Table 2. [54] pioneered the maximum product of Spacing (MPS) method for univariate distributions while Ranneby [55] developed this method to approximate the Kullback-Leibler information measure. Both researchers demonstrated that the MPS method could work in situations where the MLE method fails. ey also discovered that MPS estimators own nearly all of the MLE properties. e MPS estimator possesses almost all properties, and it gives consistent estimators with asymptotic efficiency equal to MLE estimators. Furthermore, in some cases where MLE fails, it provides consistent, asymptotically efficient estimators [56].

Maximum Product of Spacing. Cheng and Amin
e MPS estimators are regarded as values that maximize the logarithm of the sample spacing geometric. e estimated parameters μ, σ, and α.
where S n (μ, σ, α) � ln n+1 �������������� D 1 , D 2 , . . . , D n+1 μ, σ, and α estimators from the parameter μ, σ, and α could be achieved by solving the nonlinear equations as follows: where δ = the derivative of the cumulative function of the extreme distribution with respect to the estimated parameter. Reference [54] demonstrated that maximizing μ, σ, and α in MOM is as efficient as MLE. Compared to the MLE estimator, the MPS is more consistent under general conditions. e equations for estimating μ, σ, and α for various extreme value distributions using MPS are shown in Table 3.

Simulation Study
Some experimental results comparing the MOM, MLE, and MPS estimation methods were discussed in this section using a simulated study to investigate the performances of the proposed estimators. We simulated Gumbel distribution (Type I), Fréchet distribution (Type II), Weibull distribution Journal of Probability and Statistics         [57] generated N = 1,000 samples of transformed generalized exponential distribution, whereas Ramos et al. [51] chose N = 500,000 for the fitted Fréchet distribution to compare the performance of the various estimation methods. Meanwhile, Dey et al. [58] simulated N = 100,000 samples of Kumaraswamy distribution. Rodrigues [49] chose N = 10,000 to simulate the Poisson-exponential distribution with various estimation methods. Soukissian and Tsalis    [59] studied the effects of the sample size for the GEVD on the design values of wind speed. e assessment was based on a simulation study which includes each simulation is being run for 1000 random samples of each size of maxima as well as an analysis of real wind speed data. It is also reported that over 28 years in the Czech Republic, frequency analysis for two-component GEVD was applied to analyze 6-hour precipitation data from 11 stations [60]. It is critical to differentiate between two types of required sample sizes based on MOM, MPS, or MLE. As a result, we statistics was chosen as the best fit for the data. A distribution with the smallest AIC and BIC values was found to fit the data better.
AIC � 2K − 2LL, where N � sample size, K � number of parameters in the statistical model, and LL � the maximized value of the logarithmic likelihood function for the estimated model. Meanwhile, the calculation for RMSE is as follows:  EVDs using the three-parameter estimation methods considered. e estimation method which provides the smallest value of RMSE is considered as the best estimation method. Finally, a histogram with a density plot was used to compare the MPS with MLE and MOM graphically.

Estimation of Parameters.
e simulation results for all EVDs of MLE, MOM, and MPS estimation methods as well as the 95% confidence intervals are presented in Tables 4-6. e goodness-of-fit statistics for all EVDs of MLE, MOM, and MPS are presented in Tables 7 and 8 with small  and large sample sizes, respectively. Based on the tables, there are virtually no significant differences in the estimates obtained using the MLE and MPS methods. In other words, MLE and MPS variations for all EVDs were approximately 0.06% difference (for μ), 0.04% difference (for σ), and 0.02% difference (for α). e narrowest 95% CI widths are provided by MPS and MLE, respectively. Moreover, the MPS estimator provides the lowest values for MSE and bias of estimated parameters. Similarly, the values of the goodness-of-fit tests performed, Akaike's Information Criterion (AIC), the Bayesian Information Criterion (BIC), Anderson-Darling (A) test, and the Cramér-von Mises (W) test are shown in Tables 7 and 8. e estimates obtained by the MPS consistently showed lower values of goodness-of-fit statistics than those obtained by other methods for both sample sizes with different parameter values. e MOM had lower accuracy in estimating almost all of the parameters for the EVD. Nonetheless, it provided a better estimate for GEVD than the MLE method. e root mean square error (RMSE) of each parameter estimation method for all EVDs of both sample sizes is also shown in Tables 7 and 8. For both sample sizes, the MPS method has the lowest RMSE estimates for all EVDs. However, in some distributions, the difference in RMSE values for MPS and MLE estimation methods is considered almost nonexistent. RMSE values for estimates using the MOM method, on the other hand, are significantly higher for almost all distributions. is indicates that the MPS is a better fit for the EVD simulated data. As a result, various estimation methods provide a comprehensive view of the validity and performance of the estimation methods in multiple situations of extreme value analysis. Again, it is shown that MPS could be the best estimation method for fitting EVD.

Graphical Results.
A histogram is considered one of the best tools for observed data to represent the goodness-of-fit of theoretical models. It virtually provides a visual interpretation of the proposed estimation methods. Consequently, the asymptotic behavior of the proposed estimation methods is established, and their performances are investigated in the simulation study using the extreme distribution density plot. Figures 1-6 show the fitted models for all EVDs with N � 1,000 and N � 1,000,000, indicating that the MPS estimation method fitted the data well for almost all EVDs. Meanwhile, for some distributions, the MOM fitted the data to EVD with less accuracy. As illustrated in Figure 5, the MOM consistently provides a poor fit for the three-parameter Weibull distribution. is outcome is consistent with the goodnessof-fit test results for all EVDs, shown in Tables 7 and 8. As a result, the histograms show that the MPS method remains prominent for all extreme distributions.

Conclusions
is review article provides an overview of fitting EVD using MOM, MLE, and MPS. e methods' efficiency is evaluated by comparing the RMSE and several goodness-of-fit indices for two sample sizes. ree types of distributions, namely, the Gumbel, the Fréchet, and Weibull, were used to represent the distributions of extreme events. Nonetheless, determining which distribution is best suited for all extreme statistical events remains difficult. All of the examined methods can give point estimates of the GEVD parameters. However, proposing a unique parameter estimation method for all data sets and types of cases is difficult.
Based on this study, the MPS method is highly recommended regardless of the sample size because it provides better estimates for the unknown parameters and the reliability function. is review article also revealed that the majority of the related publications used MPS and other estimation methods to simulate real-life data, which offers more accurate parameter estimates. To conclude, the MPS performed better than MOM and MLE estimation methods in the majority of cases with the smallest values of RMSE and the narrowest 95% CI widths. However, the MPS provided very similar values with regard to goodness-of-fit statistics to the MLE method. erefore, the improvement of the performance of the MPS method could be taken into consideration for future studies.

Data Availability
e simulated data sets used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.