EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2461-2475 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Class of Generalized Extreme Value Distribution and Application under Alpha Power Transformation Method Pannawit Khamrot1, Natthinee Deetae2,∗ 1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok, Thailand. 2 Department of Statistics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand. Abstract. This paper presents an expansion of the generalized extreme value distribution to new distribution classes, specifically the Alpha Power Transformation Generalized Extreme Value (APTGEV) distribution. This extension is achieved by combining the Extreme Value theory and the alpha power transformation technique. We employ the maximum likelihood method in conjunction with the Newton-Raphson procedure to estimate the parameters in these proposed distributions. In the final stages of our research, we simulate these new distributions and apply them to real-world data. For this study, we have chosen extreme rainfall data from a weather station in the Si Samrong District Sukhothai Province of Thailand as our dataset. These extended distribution classes are designed to provide greater flexibility and adaptability in understanding complex data patterns, and their application to real-world data offers valuable insights into their effectiveness. 2020 Mathematics Subject Classifications: 62E10, 60G70, 62G32 Key Words and Phrases: Extremes value theory, Generalized extreme value distribution, Alpha Power Transformation Method, Return level, Maximum likelihood estimation 1. Introduction Extreme Value Theory, also known as Extreme Value Analysis (EVA), is a specialized area within statistics that focuses on the significant departures from the median in prob- ability distributions. Its purpose is to evaluate the likelihood of events surpassing any previously recorded extremities from a specified ordered sample of a particular random variable. EVA finds broad application in various fields, including meteorology, structural engineering, earth sciences, traffic forecasting, geological engineering, telecommunications, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4862 Email addresses: Pk g@rmutl.ac.th (P. Khamrot), Natthinee@psru.ac.th (N. Deetae) https://www.ejpam.com 2461 © 2023 EJPAM All rights reserved. P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2462 risk management, finance, insurance, economics, and hydrology. EVA involves estimat- ing parameters from Extreme Value (EV) distributions, commonly regarded as boundary distributions for the highest or lowest values of independent and identically distributed random variables as the sample size grows. These estimates are derived from available historical data. The practical application of EVA can be seen in numerous studies on real-world data. [1, 2, 4–7, 9, 10, 12, 13, 16, 18, 19]. A random variable X follows the Generalized Extreme Value (GEV) distribution, GEV (x;µ, σ, ξ), if its cumulative distribution function (CDF) is given by GEV (x;µ, σ, ξ) = { exp{−[1 + ξ(x− µ)/σ]−1/ξ}, ξ ̸= 0, exp{− exp[−(x− µ)/σ]}, ξ → 0, (1) which is defined in the set {x : 1 + ξ(x − µ)/σ > 0}, where µ ∈ R is a location parameter, σ > 0 is a scale parameter and ξ ∈ R is a shape parameter. The condition for a distribution possessing any of the extreme value distributions is given as follows: ξ = 0 for Gumbel distribution, ξ > 0 for Fréchet distribution and ξ < 0 for Weibull distribution. The corresponding probability density function (PDF) of GEV, gev(x;µ, σ, ξ), is then obtained as gev(x;µ, σ, ξ) = { σ−1[1 + ξ(x− µ)/σ]−(1/ξ)−1 exp{−[1 + ξ(x− µ)/σ]−1/ξ}, ξ ̸= 0, σ−1 exp[−(x− µ)/σ] exp{− exp[−(x− µ)/σ]}, ξ → 0. (2) The estimates of extreme quantiles zGEV u of the maximum distribution, known as return level, are then obtained by inverting GEV (x;µ, σ, ξ), zGEV u = µ+ σ ξ { [− log(u)−ξ − 1] } , ξ ̸= 0, µ− σ log[− log(u)], ξ → 0, (3) where u ∈ [0, 1], u = 1− T−1, and T is the number of recurrent times. In contemporary research, there is a growing interest in generalized distributions. These distributions are considered to be more adaptable as they incorporate one or more additional parameters, typically bearing a specific relationship with other distributions. The method of theoretical advancement often involves generating a new distribution de- rived from a baseline or existing distribution. This process allows for enhanced flexibility and versatility in statistical modeling, providing a broader scope for researchers to un- derstand and interpret complex data patterns. In this study, the GEV distribution is used as a baseline to seek for new classes of distribution to provide a better fit for ex- treme data. Mahdavi and Kundu [11] introduced the alpha power transformation (APT) method, which is based on adding a parameter to a family of distributions to improve their flexibility. The APT-G family’s cumulative distribution function (CDF) is defined as follows: F (x;α) =  αG(x) − 1 α− 1 , α > 0, α ̸= 1 G(x), α > 0, α = 1. (4) P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2463 The probability density function related to (4) is f(x;α) =  logα α− 1 g(x)αG(x), α > 0, α ̸= 1 g(x), α > 0, α = 1. (5) Alpha power has been applied to various distribution theories. In 2017, Nassar et al.[15] present Alpha power Weibull distribution: Properties and applications. Which is a new distribution, and two real data sets are used to illustrate the importance of the pro- posed distribution. Nadarajah et al.[14] introduced a novel three-parameter distribution and illustrated an application to an ozone data set. In 2019, Ihtisham et al.[8] studied a new distribution referred to as the Alpha-Power Pareto distribution, which is an extra parameter, and two real datasets have been considered to examine the usefulness of the proposed distribution. In 2022, Shrahili et al.[17] introduced two-parameter alpha power transformed moment exponential (APTME) distribution. To examine the practical sig- nificance of the APTME distribution and real-world datasets. The above research found that the alpha power method was extended in the distribution theory to obtain a more efficient distribution. This study proposes new variations of the GEV distribution, referred to as the Al- pha Power Transformation Generalized Extreme Value (APTGEV) distribution. The application of these proposed GEV distributions to accurate data appears encouraging, specifically with the inclusion of additional shape parameters. The structure of this paper is organized as follows. Initially, we introduce the concept of an extended distribution. Subsequently, for the proposed distributions, we elaborate on the cumulative distribution function, probability density function, and return level and provide a detailed depiction of the maximum likelihood estimates (MLEs) of all parameters. In the third part, we present a concise study that uses Monte Carlo simulation to evaluate the efficiency and consis- tency characteristics of the MLEs for the parameters of the APTGEV distribution. In the fourth section, we apply the proposed distributions to analyze rainfall data collected from weather station 373301 in Si Samrong District Sukhothai Province, Thailand, from 1987 to 2021. This analysis is followed by a goodness-of-fit test to verify the model’s accuracy. The paper concludes with a summary and interpretation of the results obtained from the study 2. Methodology In this section, we derived the forms of the cumulative distribution function, proba- bility density function, and the return level for the proposed distributions: alpha power transformation generalized extreme value distribution, including the maximum likelihood estimates (MLEs) of all parameters and confidence interval of return level. P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2464 2.1. Alpha power transformation generalized extreme value distribution A random variable X is said to have APTGEV distribution, denoted by G(µ, σ, ξ, α), with location parameter µ, shape parameter σ, and scale parameter ξ, α, if the PDF and CDF of X for x ≥ 0 and ξ ̸= 0 are given by G(x;µ, σ, ξ, α) = αexp{−[1+ξ(x−µ)/σ]−1/ξ} − 1 α− 1 , α > 0, α ̸= 1 exp{−[1 + ξ(x− µ)/σ]−1/ξ}, α > 0, α = 1. (6) The probability density function related to (6) is g(x;µ, σ, ξ, α) =  logα α− 1 σ−1[1 + ξ(x− µ)/σ]−(1/ξ)−1 exp{−[1 + ξ(x− µ)/σ]−1/ξ}αexp{−[1+ξ(x−µ)/σ]−1/ξ}, α > 0, α ̸= 1 σ−1[1 + ξ(x− µ)/σ]−(1/ξ)−1 exp{−[1 + ξ(x− µ)/σ]−1/ξ}, α > 0, α = 1, (7) and In this case ξ → 0, we have G(x;µ, σ, α) =  αexp{− exp[−(x−µ)/σ]} − 1 α− 1 , α > 0, α ̸= 1 exp{− exp[−(x− µ)/σ]}, α > 0, α = 1. (8) The probability density function (pdf) corresponding to (8) is given by g(x;µ, σ, α) =  logα α− 1 σ−1 exp[−(x− µ)/σ] exp{− exp[−(x− µ)/σ]} αexp{− exp[−(x−µ)/σ]}, α > 0, α ̸= 1 σ−1 exp[−(x− µ)/σ] exp{− exp[−(x− µ)/σ]}, α > 0, α = 1. (9) The APTGEV distribution includes three types of distribution as ξ = 0 is alpha power transformation gumbel distribution (APTGD), ξ > 0 is alpha power transformation fréchet distribution (APTFD) and ξ < 0 is alpha power transformation Weibull distribution (APTWD). Plots of the APTGEV probability density function for specific values of parameters are displayed in Figure 1. The quantile function of the APTGEV distribution for α > 0, α ̸= 1 is xp = µ+ σ ξ ((log(log(α))− log(log(p(α− 1) + 1)))−ξ − 1), ξ ̸= 0 µ− σ log(log(log(α))− log(log(p(α− 1) + 1))), ξ → 0. (10) The estimates of extreme quantiles of the maximum distribution, known as return level RT (u), are then obtained by inverting of G(x;µ, σ, α) for α > 0, α ̸= 1 is then obtained as P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2465 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 x f( x ) alpha=0.1 alpha=0.25 alpha=0.5 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .5 1 .0 1 .5 2 .0 x f( x ) alpha=0.75 alpha=0.99 alpha=1.5 Figure 1: Plots of the APTGEV probability density function as µ = 0.5, σ = 0.3, and ξ = 0.5 RT (u) = µ+ σ ξ ((log(log(α))− log(log(u(α− 1) + 1)))−ξ − 1), ξ ̸= 0 µ− σ log(log(log(α))− log(log(u(α− 1) + 1))), ξ → 0. (11) where u ∈ [0, 1], u = 1− T−1, and T is the number of recurrent times. 2.2. Distribution properties In this section, we derive the expressions for some essential properties of the APTGEV distribution. The power series can be represented by αw = ∞∑ i=0 (logα)i i! wi. (12) Hence, inserting (12) in PDF (7), then g(x;µ, σ, ξ, α) = logα α− 1 σ−1[1 + ξ(x− µ)/σ]−(1/ξ)−1 exp{−[1 + ξ(x− µ)/σ]−1/ξ} ∞∑ i=0 (logα)i i! exp{−i[1 + ξ(x− µ)/σ]−1/ξ} = ∞∑ i=0 (logα)i+1 (α− 1)i! σ−1[1 + ξ(x− µ)/σ]−(1/ξ)−1 exp{−(i+ 1)[1 + ξ(x− µ)/σ]−1/ξ} = ∞∑ i=0 Wiσ −1uξ+1e−(i+1)u where Wi = (logα)i+1 (α− 1)i! and u = [1 + ξ(x− µ)/σ]−(1/ξ). P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2466 If X has the PDF, then its kth moment, we used binomial expression can be obtained as follows. µ′ k = ∞∑ i=0 Wiσ −1 ∫ ∞ 0 xkuξ+1e−(i+1)udx = ∞∑ i=0 k∑ j,m=0 µk−j(−1)k−m−1σkk!k! (k − j)!(k −m)!j!m!ξk Wi ∫ ∞ 0 u−ξme−(i+1)udu = ∞∑ i=0 k∑ j,m=0 τi,j,m(i+ 1)(ξm−1)Γ(1− ξm), where τi,j,m = µk−j(−1)k−m−1σkk!k! (k − j)!(k −m)!j!m!ξk Wi and Γ(.) stands for the gamma function. The first four moments about zero can be found by setting k = 1, 2, 3 and 4 in µ′ k. Also, the APTGEV distribution’s moment generating function can be found as follows: Mx(t) = E(etx) = ∞∑ k=0 tk k! ∞∑ i=0 k∑ j,m=0 τi,j,m(i+ 1)(ξm−1)Γ(1− ξm). The central moments (µk) of APTGEV distribution can be obtained from µk = E(X − µ′ 1) k = k∑ n=0 (−1)n ( k n ) (µ′ 1) nµ′ k−n. Some simulations presented how the simple mean (x̄), standard deviation (SD), skew- ness values (S.V.), and kurtosis values (K.S.) of the APTGEV distribution. The result is changed for different α values, and random samples with 1,000 are generated 1,000 times by the quantile function of the APTGEV distribution, shown in Table 1 2.3. The maximum likelihood estimates (MLEs) All parameters of the APTGEV distributions are estimated from a complete set of samples using the maximum likelihood method, accompanied by the Newton-Raphson procedure. When the random sample X’s follows APTGEV distribution, the likelihood and log-likelihood functions for ξ ̸= 0 can be respectively written as L(µ, σ, ξ, α;x) = n∏ i=1 logα α− 1 σ−1[1 + ξ(xi − µ)/σ]−(1/ξ)−1 exp{−[1 + ξ(xi − µ)/σ]−1/ξ}αexp{−[1+ξ(xi−µ)/σ]−1/ξ} P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2467 Table 1: The simulation results for the APTGEV distribution with µ = 0.5, σ = 0.3 and ξ = 0.5 α Mean SD S.V. K.V. 0.01 0.5594 0.1273 -2.2132 4.6335 0.05 0.4441 0.1116 -1.9081 3.3215 0.10 0.3678 0.1020 -1.6687 2.3618 0.30 0.1556 0.0685 -1.0045 0.2559 1.50 0.5030 0.2951 -0.0348 -1.1410 3.00 0.5031 0.2949 -0.0336 -1.1448 5.00 0.5031 0.2949 -0.0364 -1.1432 10.00 0.5032 0.2947 -0.0299 -1.1497 20.00 0.5032 0.2947 -0.0321 -1.1472 40.00 0.5032 0.2946 -0.0305 -1.1506 and l(µ, σ, ξ, α;x) =n log ( logα σ(α− 1) ) − (1/ξ + 1) n∑ i=1 [1 + ξ(x− µ)/σ]− n∑ i=1 ([1 + ξ(x− µ)/σ]−1/ξ) + logα n∑ i=1 exp [−1 + ξ(x− µ)/σ]−1/ξ. When the random sample X’s are distributed as APTGEV, the respective likelihood and log-likelihood functions for ξ → 0 can be described as L(µ, σ, α;x) = n∏ i=1 logα α− 1 σ−1 exp[−(xi − µ)/σ] exp{− exp[−(xi − µ)/σ]} αexp{− exp[−(xi−µ)/σ]}, and l(µ, σ, α;x) =n log ( logα σ(α− 1) ) + n∑ i=1 (−(xi − µ)/σ)− n∑ i=1 exp(−(xi − µ)/σ) + n∑ i=1 exp(− exp(−(xi − µ)/σ)) log(α). 2.4. Confidence interval of return level The confidence interval of the return level for APTGEV distribution is performed using the Delta method as V ar(RT ) ≈ ▽Rt TV ▽ RT , where V is a covariance matrix of (µ, σ, ξ, α)t and ▽Rt T = [ ∂RT ∂µ , ∂RT ∂σ , ∂RT ∂ξ , ∂RT ∂α ]. P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2468 Where ∂RT ∂µ =1, ∂RT ∂σ =ξ−1((log(logα)− log(log(u(α− 1)) + 1))−ξ − 1), ∂RT ∂ξ =− σξ−1(log(logα)− log(log(u(α− 1)) + 1))−ξ log(log(logα) − log(log(u(α− 1)) + 1))− σξ−2(log(logα)− log(log(u(α− 1)) + 1))−ξ + σξ−2, ∂RT ∂α =− σ(log(log(α))− log(log(u(α− 1) + 1)))−ξ−1( 1 α log(α) − u (u(α− 1) + 1) log(u(α− 1) + 1) ) . The Wald approach is a well-known technique regularly employed in constructing confidence intervals for a particular parameter of interest. When performing extreme value analysis, especially when dealing with APTGEV distributions, the Wald method remains relevant. It facilitates the computation of the confidence interval for a de- sired parameter, denoted by θ. The confidence interval is determined using the equation θ±Zα/2 × √ V ar(RT ). Here, Zα/2 represents the critical value from the standard normal distribution that cuts off the upper α/2 area in the tails, and √ V ar(RT ) is the square root of the estimated variance. It’s worth noting that this estimated variance, denoted by V ar(RT ), is derived using the Delta method, a common statistical technique used for approximating the variance of a function of a random variable. 3. Simulation Study This section discusses implementing a Monte Carlo simulation study to ascertain the efficacy of Maximum Likelihood Estimates (MLEs) applied to the Asymmetric APTGEV distribution. The Inverse Cumulative Distribution Function technique produces random samples corresponding to the APTGEV distribution. This study uses the R programming language to conduct simulations and generate the resulting data. The simulation is performed N = 1,000 times on the artificial data. Multiple sample sizes n = 25, 50, 100, 500, and 1,000 are derived from the APTGEV distribution via the inverse CDF technique. Subsequently, the Newton-Raphson method is employed with the maximum likelihood estimation technique to compute the estimated parameter values. The entire process is repeated for all parameters, facilitating the calculation of the biases and Mean Squared Errors (MSEs) for the APTGEV distribution in Table 2. Upon scrutinizing the simulation results, it becomes evident that the estimated values for all parameters closely match the pre-defined ones, signifying a remarkable level of accuracy. Moreover, an intriguing pattern is discernible when analyzing the mean squared errors (MSEs) for all parameters: they consistently converge toward zero. This consistent tendency suggests an increasing precision in the estimates, a trend vividly illustrated in P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2469 Table 2. The biases of µ, σ, and ξ tend to decrease as the sample size increases while the bias of α increases. The estimated values of all four parameters in Table 2 are rather consistent across sample sizes, and Figure 2 shows a histogram of one simulated sample (for example, n=500) and its estimated density. Table 2: Simulation for APTGEV distribution for some parameter values with µ = 0.3, σ = 0.1, ξ = 0.5, α = 3 Parameter n µ σ ξ α 25 MLE 0.31030 0.13808 0.37270 2.56052 MSEs 0.00092 0.00331 0.05837 0.88913 Biase 0.01030 0.03808 -0.12730 -0.43948 50 MLE 0.30244 0.11523 0.45811 2.96645 MSEs 0.00023 0.00107 0.02139 0.38388 Biase 0.00244 0.01523 -0.04189 -0.03355 100 MLE 0.30231 0.10770 0.47998 3.10213 MSEs 0.00011 0.00052 0.01131 0.23723 Biase 0.00231 0.00770 -0.02002 0.10213 500 MLE 0.30183 0.10122 0.50090 3.21792 MSEs 0.00003 0.00008 0.00196 0.08718 Biase 0.00211 0.00144 -0.00082 0.20959 1000 MLE 0.30215 0.10047 0.50376 3.21847 MSEs 0.00002 0.00004 0.00104 0.06933 Biase 0.00204 0.00064 0.00323 0.21631 4. Application of rainfall This study applies the proposed APTGEV distributions to the highest monthly rainfall data in Sukhothai province, Thailand, during 1987 – 2021. Weather station 373301- Si Samrong District Sukhothai Province collects data from the Thailand Meteorological Department, covering an area of 6,596 square kilometers (4,122,557.5 rai). The rainy season starts around May - October, which is the period when the southwest monsoon winds. Which is hot and humid, blowing from the Indian Ocean into Thailand. The rain began to fall a lot around the middle of May onwards. The rainiest month is September, with an average rainfall of 1,259.2 millimeters per year. The temperature varies in the range of 19°C to 37°C and very rarely below 16°C or above 39°C. With 176 observations, the maximum rainfall is 176.70 mm, with a means of 57.41 mm and a standard deviation of 25.42 mm. For parameter estimation in GEV and APTGEV distributions, we utilize the maximum likelihood method accompanied by Newton–Raphson procedure. Next, we illustrate the results of parameter estimates in the GEV and APTGEV distributions in Table 3. P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2470 Density Plot z f( z ) 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 Figure 2: Plots of the APTGEV probability density function as simulated with sample size 500 Table 3: Maximum likelihood estimates, standard errors and 95% CI of all parameters in GEV and APTGEV distribution Distribution Parameter Estimate Standard error 95% CI GEV µ 44.34251 1.1199 (42.1476,46.5374) σ 12.5069 0.9905 (10.5654,14.4483) ξ 0.3744 0.0831 (0.2114,0.5373) APTGEV µ 40.5331 4.2223 (32.2574,48.8089) σ 9.79247 2.9535 (4.0036,15.5813) ξ 0.3992 0.0863 (0.2301,0.5683) α 2.9367 3.8754 (−4.6590,10.5324) In Table 3, the parameters in GEV distribution are estimated using the maximum likelihood method. The estimated values are (µ, σ, ξ) = (44.34251, 12.5069, 0.3744), with standard errors (1.1199, 0.9905, 0.0831). The approximate 95% CI for the parameters are (42.1476,46.5374) for µ, (10.5654, 14.4483) for σ, and (0.2114,0.5373) for ξ. Since the confidence interval of ξ > 0, fréchet distribution is the optimal distribution for the GEV class. Similarly, the APTGEV parameters are estimated using the maximum likelihood method. The estimated results, as shown in Table 3, are (µ, σ, ξ, α) = (40.5331, 9.79247, 0.3992, 2.9367), with standard errors (4.2223, 2.9535, 0.0863, 3.8754). The approxi- mate 95% CI for the parameters are thus (32.2574,48.8089) for µ, (4.0036,15.5813) for σ, (0.2301,0.5683) for ξ, and (−4.6590,10.5324) for α. Since zero lies within the confidence interval of ξ is contains 0, APTGD is the optimal distribution for the APTGEV class. Table 4 shows the return level estimates at different return periods (T) of 10, 20, 50 and 100 years, based on two different distributions: GEV and APTGEV. For the GEV distribution, the estimated return levels are higher as the return period P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2471 Table 4: Return level estimates (mm) at selected return periods (T) for the GEV and APTGEV distributions Distribution T Return Level 95% CI GEV 10 88.4916 (77.0835, 99.8997) 20 112.4703 (91.1340, 133.8066) 50 154.8332 (109.9808, 199.6857) 100 197.8030 (123.4002, 272.2057) APTGEV 10 88.8589 (77.1743, 100.5435) 20 113.3955 (91.4084, 135.3826) 50 157.5150 (110.6834, 204.3466 ) 100 203.1066 (124.4146, 281.7987) increases. For example, the estimated return level for a return period of 10 years is 88.4916 mm, while it is 197.8030 mm for a return period of 100 years. As a result, the lower and upper limits for confidence intervals also increase as the return period rises. Similarly, the APTGEV distribution shows an increasing trend in the estimated return levels with increasing return periods. For example, the estimated return levels for return period of 10 years and 100 years are 88.8589 mm, and 203.1066 mm, respectively. As a consequence, the lower and upper limits for CIs also increase with the return period. 4.1. Goodness-of-fit tests We will apply structured goodness-of-fit examinations to ascertain the distribution that best aligns with the data. The statistical tools we’ll utilize include the Cramér-von Mises (W ∗) and Anderson-Darling (A∗) methods delineated by Chen and Balakrishnan [3]. In general, smaller values of these statistics indicate a better fit of the distribution to the data. Let F (x; θ) be a CDF, where the form of F is known, but θ is unknown. To obtain the statistics W ∗ and A∗ as follows: W 2 = n∑ i=1 { ui − 2i− 1 2n }2 + 1 12n and A2 =− n− 1 n n∑ i=1 {(2i− 1) log(ui) + (2n+ 1− 2i) log(1− ui)}, modify W 2 into W ∗ = W 2(1 + 0.5/n) and A2 into A∗ = A2(1 + 0.75/n+ 2.25/n2), where ui = Φ{(yi − ȳ)/sy}, ȳ = (1/n) ∑n i=1 yi, s 2 y = (n − 1)−1 ∑n i=1(yi − ȳ)2, yi = Φ−1(vi), vi = F (xi; θ̂), the xi’s are in ascending order, Φ(·) is the standard normal CDF and Φ−1(·) denotes its inverse. The values of W ∗ and A∗ for GEV and APTGEV distribution are given in Table 5 for goodness-of-fit tests on rainfall data. Based on the values of W ∗ and A∗, the proposed P. Khamrot, N. Deetae / Eur. J. Pure Appl. Math, 16 (4) (2023), 2461-2475 2472 Table 5: Goodness-of-fit tests for the GEV and APTGEV distributions Distribution W ∗ p-value A∗ p-value GEV 0.0454 0.9040 0.3436 0.9020 APTGEV 0.0414 0.9262 0.3247 0.9184 APTGEV distribution fits the data better than GEV distribution. Therefore, it could be considered as an alternative to other distributions in the literature for positive real data. Figure 3 presents the diagnostic plots for the APTGEV distributions. Most points in the probability and quantile plots closely align with the unit diagonal, indicating that the APTGEV distribution functions offer reliable fits. This aligns with the results from the Cramér-von Mises (W ∗) and Anderson-Darling (A∗) statistical tests. Furthermore, the density plot shows a strong concurrence between the established APTGEV distribution function and the empirical density. 5. Conclusions This paper introduces a novel extreme value distribution, the Alpha Power Transfor- mation generalized extreme value (APTGEV) distribution. Our research methodology elaborates on the maximum likelihood estimation for all parameters, quantile function, moment generating function, and returns levels of GEV and APTGVE distribution, in- cluding their confidence intervals. Owing to the complexity of likelihood functions, we employ the Newton–Raphson method to derive the estimated values for all parameters. Through Monte Carlo experiments in a simulation study, we examine the behavior of these maximum likelihood estimates, which closely align with their initial assigned values. All parameters’ mean squared errors (MSEs) converge toward zero. Notably, the biases for parameters µ, σ, and ξ diminish with an increase in sample size, while the bias for α rises. Yet, across different sample sizes, the estimated values for all four parameters remain rel- atively consistent. Regarding practical applications, we apply the proposed distributions to rainfall data. The results reveal a commendable fit per the Cramér-von Mises and Anderson-Darling goodness-of-fit tests. Thus, these proposed distributions demonstrate their effectiveness in fitting extreme data sets. We anticipate this generalization will have extensive applicability across disciplines, including statistics, mathematics, biology, envi- ronmental science, engineering, and more. Moreover, our study provides valuable insights for meteorologists regarding the behavior of high rainfall events within the context of ex- treme value theory. Both governmental and non-governmental entities can leverage these insights to make informed decisions and devise contingency plans in preparation for the effects of extreme rainfall events. Such measures can aid in early warning systems, food security, poverty reduction, and disaster or risk management, among other areas. Beyond its practical applications, our study also enriches theoretical knowledge, expands the scope of extreme value theory, and makes other theories more efficient. REFERENCES 2473 Acknowledgements This work is a partially supported dataset from the Thailand Meteorological Depart- ment. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Probability Plot empirical m o d e l 50 100 150 5 0 1 0 0 1 5 0 2 0 0 2 5 0 Quantile Plot ordered m o d e l. q u a n ti le Density Plot z f( z ) 50 100 150 0 .0 0 0 0 .0 1 0 0 .0 2 0 0 .0 3 0 0 200 400 600 800 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 95% CI of Return Level Plot Return Period R e tu rn L e ve l Figure 3: Diagnostic plots for APTGEV distribution, applying to rainfall data in Phitsanulok province, Thailand References [1] S. El Adlouni, T. B. M. J. Ouarda, X. Zhang, R. Roy, and B. Bobee. Generalized max- imum likelihood estimators for the nonstationary generalized extreme value model. Water Resources Research, 43:1–13, 2007. [2] P. Busababodhin and A. Kaewmun. Extreme values statistics. The Journal of King Mongkut’s University of Technology North Bangkok, 25(2):55–65, 2015. REFERENCES 2474 [3] G. 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