Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 8, No. 2, April 2023 Research Paper Extension of Exponential Pareto Distribution with the Order Statistics: Some Properties and Application to Lifetime Datasets Adewunmi Olaniran Adeyemi1*, Ismail Adedeji Adeleke2, Eno Emmanuella Akarawak1 1Department of Statistics, Faculty of Science, University of Lagos, Lagos, 101017, Nigeria2Department of Actuarial Science and Insurance, University of Lagos, Lagos, 101017, Nigeria *Corresponding author: adewunyemi@yahoo.com AbstractThe Exponential Pareto (EP) model has been extended by applied and theoretical statisticians for wider applications and newknowledge using different techniques but the Weibull-X technique has not been considered. This article proposed a new extension ofthe EP model called the Weibull-Exponential Pareto (WEP) distribution to provide better modeling that fits real-life datasets and toexplore the statistical theory of order statistics from the proposed distribution. Statistical properties investigated include the Shannonand Renyi entropies; the moments and moment generating function. Distribution of order statistics and the moment of orderstatistics were derived including the mean and variance of order statistics. WEP distribution has unimodal, decreasing, and increasingfailure rates; and it can be negatively or positively skewed and approximately symmetric with the potential for fitting platykurtic,mesokurtic, and leptokurtic lifetime data. The parameters of the distribution were estimated using the method of maximum likelihoodestimation (MLE), which was examined for consistency through a simulation study. The performance of the proposed distributionwas investigated by application to flood peaks exceedances and some lifetime datasets from engineering. The results from dataanalysis using the R-software revealed that the WEP distribution has the potential to provide a superior model that fits the three datasets better than some notable existing distributions and previous extensions of the EP model in the literature. The statistical propertyof order statistics extended in the study established some important results that characterized some notable lifetime distributions inthe literature. KeywordsExponential Pareto Model, Weibull-X Technique, Weibull-Exponential Pareto Distribution, Shannon and Renyi Entropies, Distributionof Order Statistics, Moment of Order Statistics Received: 9 November 2022, Accepted: 6 February 2023 https://doi.org/10.26554/sti.2023.8.2.265-279 1. INTRODUCTION Statisticians and researchers in general are motivated by burn- ing desires to discover new distributions that are adequate and more flexible in terms of application to real-life problems. Ex- tension of classical distributions and some other existing mod- els have been suggested and successfully implemented in the statistical literature using various techniques. Previous efforts towards the actualization of the objectives are contained in Lee et al. (2007) using the beta-generator technique to extend the Weibull distribution, the T-X family of distribution was introduced by Alzaatreh et al. (2013b) and by taking T to be a Weibull distribution random variable, the Weibull-X was defined by Alzaatreh et al. (2013b) as a sub-family of the T-X family. Several important distributions have been proposed using the Weibull-X technique including the Weibull-Pareto model by (Alzaatreh et al., 2013a) . The Weibull-Rayleigh dis- tribution was developed by Akarawak et al. (2013) and later by Ahmad et al. (2017) with different motives and diverse applications when X follows the Rayleigh random variable. Al Kadim and Boshi (2013) defined and studied the Expo- nential Pareto (EP) distribution with the cumulative distribu- tion function (CDF) and probability density function (PDF) defined respectively as F(x) = 1 − e− 𝛽 ( x k ) \ (1) f (x) = 𝛽 \ k ( x k )\−1 e− 𝛽 ( x k ) \ ; 𝛽 , k, \ > 0 > 0;x > 0 (2) 𝛽 ,k,\ are the parameters of the distribution The EP distri- bution has further gained extensive studies with applications https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2023.8.2.265-279&domain=pdf https://doi.org/10.26554/sti.2023.8.2.265-279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 from Luguterah and Nasiru (2015) , using the quadratic rank transmutation map (QRTM) to develop the Transmuted Ex- ponential Pareto (TEP) distribution, the beta-G framework was used by Aryal (2019) and later by Rashwan and Kamel (2020) for the construction of Beta Exponential Pareto (BEP) model. Kumaraswamy Exponential Pareto (KEP) distribution was introduced by Elbatal and Aryal (2017) using the Kum-G technique and most recently, the Gompertz-G technique was explored for developing the Gompertz Exponential Pareto dis- tribution by (Adeyemi et al., 2021) . Dikko and Faisal (2017) proposed the generalized exponential Weibull (GEW) distri- bution and the Topp Leone Weibull distribution (TLWD) was introduced by (Tuoyo et al., 2021) . In another dimension, the new Weibull Pareto distribution (NWP) was developed by Nasiru and Luguterah (2015) ; and thereafter, with the aid of the QRTM, Tahir and Akhter (2018) extended the NWP to produce the Transmuted new Weibull Pareto (TNWP) and most recently, Aljuhani et al. (2022) in- troduced the Alpha Power Exponentiated New Wuibull Pareto distribution from the NWP model and Hassan et al. (2022) developed the Kumaraswamy extended Exponential (KwEE). Nevertheless, instances abound where these existing distribu- tions have not been able to explain some of the real-life prob- lems adequately through the analysis of their corresponding datasets, making this study gap a challenge that is constantly been addressed by researchers. This current study is aimed at exploring the versatility of the Weibull distribution as a generator by using the Weibull- X technique to extend the EP model. The proposed model called the Weibull Exponential Pareto (WEP) distribution is ex- pected to provide more flexibility for addressing various forms of kurtosis and skewness associated with real-life datasets de- scribing some of the random events in our environments. The remaining part of the study is organized as follows; Section 2 is devoted to describing and developing the proposed distribution with the sub-models and some of the statistical properties. Sec- tion 3 discussed the procedure for the estimation of parameters and simulation study. Applications to three real-life datasets were carried out to assess the importance of the distribution in Section 3. Section 4 concludes the work and Section 5 for acknowledgment. 2. EXPERIMENTAL SECTION This section is used for the design of the new distributions from some existing resources in the statistical literatures; properties of the WEP distribution constructed are also investigated. 2.1 Materials and Method A new method of generating continuous distribution proposed by Alzaatreh et al. (2013b) has the CDF for the T-X class of distribution defined as G(x) = ∫ −log(1−F (x)) 0 r(t)dt = R{−log(1 − F(x))} (3) Where R(t) the CDF of a non-negative continuous random variable is T defined on [0,∞) and F(x) is the CDF of a random variable X. The PDF associated with Equation (3) is given by, G(x) = f (x) 1 − F(x) r{−log(1 − F(x))} (4) Where r(t) is the PDF of random variable T and the deriva- tive of R(t). Let T be a random variable from the Weibull distribution with parameters 𝛼 and 𝛾 having the CDF given by, R(t) = 1 − exp ( − ( t 𝛾 )𝛼 ) ; 𝛼, 𝛾 > 0, t > 0 (5) The CDF of the Weibull-X family is derived by inserting t=-log(1-F(x)) into Equation (5) to obtain Alzaatreh and Ghosh (2015) given by G(x) = 1 − exp ( − ( −log(1 − F(x)) 𝛾 )𝛼 ) (6) And the PDF can be derived by taking the first derivative of Equation (6) to get g(x) = 𝛼 𝛾 f (x) 1 − F(x) ( −log(1 − F(x)) 𝛾 )𝛼−1 exp( − ( −log(1 − F(x)) 𝛾 )𝛼 ) (7) 2.2 The New Extension of Exponential Pareto Distribution The CDF of the proposed lifetime distribution called Weibull Exponential Pareto (WEP) distribution when X follows the EP distribution in Equation (1) is developed as; G1(x) = 1 − exp ( − ( 𝛽 ( x k )\ 𝛾 )𝛼 ) (8) Then by replacing 𝛽 ⁄𝛾 with _ , the CDF in Equation (8) can be written as G(x) = 1 − exp ( − ( _ ( x k )\ )𝛼 ) (9) The derivative of Equation (9) is the corresponding PDF of the Weibull Exponential Pareto (WEP) distribution given by g(x) = 𝛼_ \ k ( x k )\−1 ( _ ( x k )\ )𝛼−1 exp ( − ( _ ( x k )\ )𝛼 ) ; 𝛼, _ , \ , k > 0;x > 0 (10) A random variable X that follows the Weibull Exponential Pareto distribution with parameters 𝛼,_ ,\, and k is character- ized with the density function g(x) in Equation (10) and is denoted by WEP (𝛼,_ ,\,k). © 2023 The Authors. Page 266 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 2.3 Sub-Models of WEP Distribution The important sub-models derived from the proposed distri- bution are presented in Table 1. Plots of the shapes for the CDF and PDF of the distribution for some parameter values are displayed in Figure 1. Figure 2 provides the visual view of the shapes of the hazard rate function of the proposed distribution for some values of the parameter (𝛼,_ ,\,k). Figure 1. Plots of the CDFs and PDFs of WEP Distributions for Some Values of the Parameters Figure 2. Plots of Hazard Rate Functions of WEP Distributions for Some Values of the Parameters 2.4 Properties of the Weibull Exponential Pareto Distribu- tion Some of the properties of the distribution are discussed in this section 2.4.1 The Reliability and the Hazard Rate Function The reliability function is defined in similar works including Adeyemi et al. (2021) and is given by S(x) = 1 −G(x) = exp ( − ( _ ( x k )\ )𝛼 ) (11) The hazard rate function is derived from Equation (9) and Equation (10) and is given by h(x) = g(x) 1 −G(x) = 𝛼_ \ k ( x k )\−1 ( _ ( x k )\ )𝛼−1 (12) 2.4.2 Asymptotic Behavior of WEP The Asymptotic properties of the proposed model are inves- tigated by taking limits of the density function, CDF, and the hazard rate function as x→∞ and as x→0. Proposition 1: The limit of the WEP density function as x→∞ is 0 and as x→0 is; lim x→0 g(x) =  0 , 𝛼\ > 1 _ k , 𝛼\ = 1 ∞ , 𝛼\ < 1 Proof: lim x→∞ g(x) = lim x→∞  𝛼_ \ k ( x k )\−1 {_ ( xk )\ }𝛼−1 exp ( −{_ ( x k )\ }𝛼 )  = 0 lim x→0 g(x) = lim x→0 𝛼_ \ k ( x k )\−1 {_ ( x k )\ }𝛼−1 exp ( −{_ ( x k )\ }𝛼 ) The limit of exp ( − { _ ( x k )\ }𝛼 ) as x→0 is 1, We now have a situation where as x→0 when \>1, 𝛼\>1 it becomes 0, when \<1, 𝛼\<1 it becomes ∞ ; when 𝛼=\=1 we have limx→0 _ k exp (-{_ ( x k )}) which reduces to constant _k . This completes the proof. Proposition 2: The limit of WEP cdf as x→∞ is 1 and as x→0 is 0 Proof: lim x→∞ G(x) = lim x→∞ [ 1 − exp ( −{_ ( x k )\ }𝛼 ) ] = 1 − 0 = 1 lim x→0 G(x) = lim x→0 [ 1 − exp ( −{_ ( x k )\ }𝛼 ) ] = 1 − 1 = 0 Proposition 3: The limit of the WEP hazard rate function as x→∞ is ∞ and as →0 is given by 0 , 𝛼\ > 1 _ k , 𝛼\ = 1 ∞ , 𝛼\ < 1 Proof: The hazard rate function is defined as, h(x) = 𝛼_ \ k ( x k )\−1 { _ ( x k )\ }𝛼−1 lim x→∞ h(x) = ∞ 0 = ∞ when 𝛼\ > 1; lim x→∞ h(x) = ∞ 0 = ∞ when 𝛼\ < 1. The asymptotic limits when 𝛼\ = 1 is also lim x→∞ h(x) = _ k ( 1 0 ) = ∞. © 2023 The Authors. Page 267 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 Table 1. Sub-Models of WEP Distribution 𝛼 _ \ k Reduced Model Author/References 𝛼 12𝛽 2 k Weibull-Rayleigh Ahmad et al. (2017) 𝛼 1p 2 1 Weibull-Rayleigh Akarawak et al. (2017) 1 _ \ k Exponential Pareto Al Kadim and Boshi (2013) 2 1 8𝜎2 2 𝛽 Rayleigh Rayleigh Ateeq et al. (2019) 1 _∗2 2 k Exponential Rayleigh New 1 1 _ k Exponential Exponential New 1 12 2 k Rayleigh Rayleigh (1896) 1 12 \ k Weibull Weibull (1951) Asymptotes of hazard rate as x→0 lim x→0 h(x) = 0 1 = 0 when 𝛼\ > 1; lim x→0 h(x) = ∞ 1 = ∞ when 𝛼\ < 1. The asymptotic limits when 𝛼\ = 1 is also lim x→0 h(x) = _ k , the proof is completed. Proposition 4: Let g(x) be the PDF and h(x) the hazard rate function of WEP distribution then, as x→0 we have g(0)=h(0). Proof: Combining results from propositions (1) and (3) estab- lished the proof and is given by, lim x→0 h(x) =  0 , 𝛼\ > 1 _ k , 𝛼\ = 1 ∞ , 𝛼\ < 1 = lim x→0 g(x) 2.4.3 Quantile Function, Simulation, and Median Let U be a uniform random variable on the interval (0, 1); the quantile function is defined by Q(u)=G−1 (x), and the WEP distribution has the quantile function derived and presented as; Q(u) = k { 1 _ [ −log(1 − u) 1 𝛼 ] } 1 \ (13) Let X be a random variable from the Weibull Exponential Pareto distribution, simulation can be done through the inverse transformation of the variable using uniform interval U (0, 1) and the random variable X taking as, X = k { 1 _ [ −log(1 − u) 1 𝛼 ] } 1 \ (14) The median of the WEP can be derived by substituting u=0.5 in Equation (14) to get median = k { 1 _ [ −log(0.5) 1 𝛼 ] } 1 \ (15) 2.4.4 Moments of WEP Distribution Theorem 1: Let X be a continuous random variable from the WEP distribution with density function g(x), then the rth moment about the origin is given by; ` (r) = kr ( 1 _ )r/\ Γ ( r 𝛼\ + 1 ) (16) Proof : The moment of X is defined as E(xr)= ∫ ∞ −∞ x r g(x)dx E(xr) = ∫ ∞ 0 xr 𝛼_ \ k ( x k )\−1 { _ ( x k )\ }𝛼−1 exp ( − { _ ( x k )\ }𝛼 ) dx (17) Let u = _ ( x k )\ which implies that x = k ( u _ )1/\ then substitute new variables so that du dx = _ \ k ( x k )\−1 and du = _ \ k ( x k )\−1 dx E(xr) = ∫ ∞ 0 ( k ( u _ )1/\ )r 𝛼_ \ k ( x k )\−1 u𝛼−1exp(−u𝛼) 1 _ \ k ( x k )\−1du, = 𝛼kr ( 1 _ )r/\ ∫ ∞ 0 u r \ +𝛼−1(−u𝛼)du, let y = u𝛼 , so that u = y1/𝛼 hence dy du = 𝛼u𝛼−1 = 𝛼(y1/𝛼)𝛼−1 © 2023 The Authors. Page 268 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 E(xr) =𝛼kr ( 1 _ )r/\ ∫ ∞ 0 (y1/𝛼) r \ +𝛼−1exp(−y) 1 𝛼(y1/𝛼)𝛼−1 dy E(xr) = kr ( 1 _ )r/\ ∫ ∞ 0 y r 𝛼\ exp(−y)dy; Using ∫ ∞ 0 ymexp(−y)dy = Γ(m + 1)completes the proof. The first four raw moments can be obtained from Equation (16) as follows ` ′ 1 = k ( 1 𝛼 )1/\ Γ ( 1 𝛼\ + 1 ) ; ` ′ 2 = k 2 ( 1 𝛼 )2/\ Γ ( 2 𝛼\ + 1 ) ; ` ′ 3 = k 3 ( 1 𝛼 )3/\ Γ ( 3 𝛼\ + 1 ) ; ` ′ 4 = k 4 ( 1 𝛼 )4/\ Γ ( 4 𝛼\ + 1 ) ; 2.4.5 The Mean, Variance, Skewness, and Kurtosis of WEP Distribution The mean is the first moment about the origin when r = 1, and is derived as E(X) = `′1 = k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) (18) The variance is obtained as Variance(X) = `′2 − [` ′ 1] 2 = k2 { ( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) − (( 1 _ )1/\ Γ ( 2 𝛼\ + 1 ))2 } (19) The measures of skewness and kurtosis denoted by Skew and Kurt respectively are obtained using the first four raw moments in sub-section 2.4.4 as follows; Skew = `′3 − 3` ′ 2 ` + 2` 3 ( `′2 − ` 2)3/2 = ©­­­« k3 ( 1 _ )3/\ Γ ( 3 𝛼\ + 1 ) −3k2 ( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) +2 [ k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 )]3 ª®®®¬( k2 ( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) − [ k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 )2]3/2 ) (20) Kurt = `′4 − 4` ′ 3 ` + 6`2 ` 2 − 3`4 ( `′2 − ` 2)2 = ©­­­­­­« k4 ( 1 _ )4/\ Γ ( 4 𝛼\ + 1 ) − 4k3 ( 1 _ )3/\ Γ ( 3 𝛼\ + 1 ) k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) + 6k2 ( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) [ k ( 1 𝛼\ )1/\ Γ ( 1 𝛼\ + 1 )]2 − 3 [ k ( 1 𝛼\ )1/\ Γ ( 1 𝛼\ + 1 )]4 ª®®®®®®¬( k2 ( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) − [ k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 )2]3/2 ) (21) Proposition 5: Let X be a random variable from the WEP distribution, the coefficient of variance denoted CV is given by CV = √√{( 1 _ )2/\ Γ ( 2 𝛼\ + 1 ) − (( 1 _ )1/\ Γ ( 2 𝛼\ + 1 ))2} ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) (22) Proof The coefficient of variance is defined as CV = 2 √ variance E(x) By using Equations (18) and (19), the result is obtained. 2.4.6 Moment Generating Function of WEP Distribution Theorem 2: Let X be a Weibull Exponential Pareto random variable with probability density function g(x), the moment generating function of X denoted Mx (t) is given by ∞∑︁ i=0 ti i! ki ( 1 _ )i/\ Γ ( i 𝛼\ + 1 ) (23) Proof: The moment generating function for a continuous random variable is defined Alzaatreh et al. (2013a) as; Mx(t) = E(etx) = ∫ ∞ 0 etxg(x)dx ∫ ∞ 0 etx 𝛼_ \ k ( x k )\−1 { _ ( x k )\ }𝛼−1 exp ( − { _ ( x k )\ }𝛼 ) dx etx = ∞∑︁ i=0 ( 1 + tixi i! ) = ∞∑︁ i=0 tixi i! © 2023 The Authors. Page 269 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 E(etx) == ∞∑︁ i=0 ti i! E(xi) By substituting E(xi) in Equation (16), we shall obtain Mx(t) = ∞∑︁ i=0 ti i! E(xi) = ∞∑︁ i=0 ti i! ki ( 1 _ )i/\ Γ ( i 𝛼\ + 1 ) 2.4.7 Relationship between WEP and the Weibull-Pareto (WPD) Distribution The relationship between the WEP and WPD is established as follows: Theorem 3: Let Y be a random variable that follows the Weibull-Pareto distribution with parameter (𝛼,_ ,k) defined and studied by (Alzaatreh et al., 2013b) , then the random variable x=k(log ( y k ) ) 1/\ follows the WEP distribution with pa- rameters (𝛼,_ ,\,k). Proof: Given thatY follows the Weibull-Pareto distribution, then Y∼(𝛼,_ ,k), required to show that X∼(𝛼,_ ,\,k). The CDF is given by Alzaatreh et al. (2013b) as 1 − exp ( − { _ log ( y k )}𝛼 ) (24) Required to show that X∼(𝛼,_ ,\,k), X = k ( log ( Y k ))1/\ By transformation of variable, (log(Yk ))=( X k ) \ , this implies that (Yk ) = e ( Xk ) \ and the ran- dom variable Y = ke( X k ) \ By substituting Y into Equation (24), the CDF of WEP distribution in Equation (9) is obtained. 2.4.8 Shannon Entropy of Weibull Exponential Pareto Dis- tribution Shannon (2001) defined entropy as the measure of the level of variation of uncertainty associated with a random variable. If a random variable T follows the Weibull distribution with param- eters c and 𝛾, the Shannon entropy for the Weibull distribution by Song (2001) is given by [T = 𝜗 ( 1 − 1 c ) − log ( c 𝛾 ) + 1 (25) The Shannon entropy for the Weibull-X family of distri- bution by Alzaatreh et al. (2013b) is given by [x = −{logf (F−1(1 − e−T))} − 𝛾Γ ( 1 + 1 c ) + 𝜗 ( 1 − 1 c ) − log ( c 𝛾 ) + 1 (26) Where 𝜗 the Euler’s constant in both equations. The mean of Weibull distribution for the random variable T is `T = 𝛾Γ ( 1 + 1 c ) (27) Theorem 4: Let X be a random variable from the WEP distribution, then the Shannon entropy [xfor the WEP distri- bution is given by k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) − Γ ( 1 + 1 𝛼 ) + 𝜗 ( 1 − 1 𝛼 ) − log(𝛼) + 1 (28) Proof: By definition, Shannon entropy [X is the expecta- tion of the negative logarithm of the density function g(x). The expectation of the PDF of the Weibull-X family in Equation (4) is given by E[−log{g(x)}] = E [ −log ( f (x) 1 − F(x) r{−log(1 − F(x))} )] E[−log{f (x)} + log(1 − F(x)) − log[r{−log(1 − F(x))}]] E[−log( f (x))] + E[log(1 − F(x))]+ E[−log[r{−log(1 − F(x))}]] By variable transformation, if T= -log (1-F(x)), then (1- F(x))=e−T , [x = E[−log{g(x)}] = E[−log(x)] − E[T]+ E[−log[r(t)]] = `x − `T + [T Where `x = The mean of WEP distribution `T = The mean of Weibull distribution [T = The Shannon entropy of Weibull distribution [X= The Shannon entropy of the Weibull-X distribution By substituting for `x which is the mean of WEP in Equa- tion (18) and also the values of [T and `T in Equations (19) and (27) respectively, the Shannon entropy [X of the WEP distribution is obtained as [x = k ( 1 _ )1/\ Γ ( 1 𝛼\ + 1 ) − Γ ( 1 + 1 𝛼 ) + 𝜗 ( 1 − 1 𝛼 ) − log(𝛼) + 1 © 2023 The Authors. Page 270 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 2.4.9 Renyi Entropy of Weibull Exponential Pareto Distri- bution The Renyi entropy of a continuous random variable X with PDF g(x) is defined by Rényi (1961) as the measure of uncer- tainty associated with X, and the Renyi entropy of X is defined by ∥R(X) = 1 1 − 𝛿 log[∥(𝛿)] (29) Where ∥(𝛿) = ∫ ∞ −∞ .f 𝛿 (x)dx, 𝛿 > 0 and 𝛿 ≠ 1 (30) The Renyi entropy of a random variable X that follows the WEP distribution is derived by substituting the PDF of WEP into Equation (30) ∥(𝛿) = ∫ ∞ ∞ ( 𝛼_ \ k )𝛿 ( x k )𝛿(\−1) { _ ( x k )\ }𝛿(𝛼−1) exp ( − { 𝛿_ ( x k )\ }𝛼 ) dx Let u = 𝛿_ ( x k )\ and obtain x = k ( u 𝛿_ )1/\ du dx = 𝛿_ \ k ( x k )\−1 du = 𝛿_ \ k ( x k )\−1 dx ∥(𝛿) = ∫ 0 ∞ x\𝛿𝛼−𝛿 (𝛼_ \)𝛿 _ 𝛿𝛼−𝛿 ( 1k ) \𝛿𝛼 exp(−u𝛼) 𝛿_ \ k ( 1 k ) \−1 ( k( u 𝛿_ )1/\ )\−1du = ∫ 0 ∞ ( 1 _ )− 𝛿 \ + 1 \ (u 𝛿 )𝛿𝛼− 𝛿 \ + 1 \ 𝛼 𝛿 ( \ k )𝛿−1 exp(−u𝛼)(u)−1du Let y = u𝛼 , dy du = 𝛼u𝛼−1 = 𝛼(y1/𝛼)𝛼−1 ∥(𝛿) = ∫ 0 ∞ ( 1 _ )− 𝛿 \ + 1 \ ( y1/_ 𝛿 )𝛿𝛼− 𝛿 \ + 1 \ 𝛼 𝛿 ( \ k )𝛿−1 exp(−y)(y1/𝛼)−1 𝛼(y1/𝛼)𝛼−1 dy = ∫ 0 ∞ (_) 1−𝛿 \ ( 1 𝛿 )𝛿𝛼− 𝛿 \ + 1 \ ( 𝛼\ k )𝛿−1 exp(−y)y𝛿− 𝛿 𝛼𝛿 + 1 𝛼\ −1dy ∥(𝛿) = (_) 1−𝛿 \ ( 1 𝛿 ) 𝛿𝛼\−𝛿+1 \ ( 𝛼\ k )𝛿−1 Γ ( 𝛿𝛼\ − 𝛿 + 1 𝛼\ ) (31) Substitute the quantity in Equation (31) into Equation (29) to get the desired result given by ∥R(X) = 1 1 − 𝛿 log[∥(𝛿)] = 1 1 − 𝛿 log [ (_) 1−𝛿 \ ( 1 𝛿 ) 𝛿𝛼\−𝛿+1 \ ( 𝛼\ k )𝛿−1 Γ ( 𝛿𝛼\ − 𝛿 + 1 𝛼\ )] 2.4.10 Distribution of the Order Statistics of WEP Distri- bution Let X1,X2,. . . ,xn be a random sample of size n from the WEP distribution with the CDF and PDF given as G(x) and g(x) respectively, if X(1) , X(2) ,....,X(n) is the order statistics of the random sample, the PDF of the rth order statistics defined by David and Nagaraja (2003) is given by f xr:n(x) = n (r − 1)(n − r)! G(x)r−1(1 −G(x))n−rG′(x) (32) Substitute the CDF and PDF of the Weibull Exponen- tial Pareto distribution defined in Equations (9) and (10) into Equation (32) to get f xr;n(x) =  n (r−1) (n−r)! [ 1 − exp ( − { _ ( x k )\ }𝛼 )]r−1[ exp ( − { _ ( x k )\ }𝛼 )]n−r+1 X 𝛼_ \k ( x k )\−1 { _ ( x k )\ }𝛼−1 (33) The first-order statistics is obtained from Equation (33) when r=1 and is given by f x1:n(x) = [ exp ( − { _ ( x k )\ }𝛼 )]n n𝛼_ \ k ( x k )\−1 { _ ( x k )\ }𝛼−1 (34) The maximum order statistics is derived from Equation (33) when r=n and is given by f xn;n(x) =  [ 1 − exp ( − { _ ( x k )\ }𝛼 )]n−1[ exp ( − { _ ( x k )\ }𝛼 )] X n𝛼_ \k ( x k )\−1 { _ ( x k )\ }𝛼−1 (35) © 2023 The Authors. Page 271 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 2.4.11 Moments, Mean, and Variance of Order Statistics of WEP Distribution The moment of order statistics including the mean and variance of order statistics are derived here Theorem 5: Let X1:n, X2:n, . . . , Xn:n be the order statistics of the random sample from the WEP distribution of the density function fxr;n(x), derived in Equation (33), then the sth moment of the rth order statistics is given by ` s r.n =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) ks ( 1 _ )s/\ X ( 1 m ) ( s 𝛼\ +1) Γ ( s 𝛼\ + 1 ) (36) Proof: The sth moment of X(r:n) is defined as E(Xsr:n) = ∫ ∞ −∞ xs f xr;n(x)dx (37) Equation (33) can be expressed by series expansion as  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) 𝛼_ \ k ( x k )\−1 X { _ ( x k )\ }𝛼−1 exp ( − { m_ ( x k )\ }𝛼 ) (38) Where m=n-r+i+1, and Cr:n = n (r−1)!(n−r)! Substitute Equation (38) into (37) to obtain  Cr:n ∑r−1 i=0 (−1) i ( r − 1 i ) ∫ ∞ 0 xs 𝛼_ \k ( x k )\−1 X { _ ( x k )\ }𝛼−1 exp ( − { m_ ( x k )\ }𝛼 ) dx (39) By following the steps in subsection 2.4.4, Theorem 1, the solution to the integral in Equation (39) is obtained as ks ( 1 _ )s/\ ( 1 m )( s𝛼\ +1) Γ ( s 𝛼\ + 1 ) (40) And the sth moment of the rth order statistics of WEP dis- tribution is given by Cr:n r−1∑︁ i=0 (−1)i ( r − 1 i ) ks ( 1 _ )s/\ ( 1 m )( s𝛼\ +1) Γ ( s 𝛼\ + 1 ) Corollary 1: The mean of order statistics of the WEP dis- tribution is obtained as `r;nWEP =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) k ( 1 _ )1/\ X ( 1 m )( 1𝛼\ +1) Γ ( 1 𝛼\ + 1 ) (41) Corollary 2: Let 𝜎sr:n(WEP) denote the variance of order statistics of the WEP distribution, then the explicit expression for the variance is given by 𝜎 s r:nWEP =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) k2( 1 _ )2/\ ( 1 m )( 2𝛼\ +1) Γ ( 2 𝛼\ + 1 ) − [ n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) k( 1 _ )1/\ ( 1 m )( 1𝛼\ +1) Γ ( 1 𝛼\ + 1 )]2 (42) Proof: The variance is proved using the relation; 𝜎 s r:nWEP = ` 2 r:n − ( `r:nWEP) 2 Using Equation (36), `2r:n is obtained as ` 2 r:n = n (r − 1)!(n − r)! r−1∑︁ i=0 (−1)i ( r − 1 i ) k2 ( 1 _ )2/\ ( 1 m )( 2𝛼\ +1) Γ ( 2 𝛼\ + 1 ) Using Equation (41), (`r;nWEP ) 2 is obtained as ( `r:nWEP)2 = [ n (r − 1)!(n − r)! r−1∑︁ i=0 (−1)i ( r − 1 i ) k ( 1 _ )1/\ ( 1 m )( 1𝛼\ +1) Γ ( 1 𝛼\ + 1 )]2 3. RESULTS AND DISCUSSION 3.1 GeneralizationofPropertiesofOrderStatistics forSome Lifetime Distribution The mean, variance, skewness, kurtosis, and some other im- portant statistical properties can be derived for some lifetime distributions using the theory and application of order statis- tics. This sub-section generalized some few properties of order statistics for some lifetime distribution as follows: © 2023 The Authors. Page 272 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 3.1.1 ExponentialParetoDistribution(AlKadimandBoshi, 2013) Corollary 3: If 𝛼=1, in Equation (36) the result for sth moment of the rth order statistics of the EP distribution is given by ` s r:nEP =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) ks ( 1 _ )s/\ X ( 1 m )( s\ +1) Γ ( s \ + 1 ) (43) The mean of order statistics of EP distribution is derived and given by `r:nEP =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) k ( 1 _ )1/\ X ( 1 m )( 1\ +1) Γ ( 1 \ + 1 ) (44) The mean of the minimum X(1:n) order statistics of EP distri- bution is given by `1:nEP = nk ( 1 _ )1/\ Γ ( 1 \ + 1 ) (45) The mean of the maximum X(n:n) order statistics of EP distri- bution is given by `n:nEP = n−1∑︁ i=0 (−1)i ( n − 1 i ) nk ( 1 _ )1/\ ( 1 i + 1 )( 1\ +1) Γ ( 1 \ + 1 ) (46) Corollary 4: If r=n=1 and 𝛼=1, the result for a central moment about the origin and the mean for the EP distribution Al Kadim and Boshi (2013) is obtained from this study as ` ′ s = k s ( 1 _ )s/\ Γ ( s \ + 1 ) (47) The mean of a random variable X from the EP distribution is E(X) = k ( 1 _ )1/\ Γ ( 1 \ + 1 ) (48) 3.1.2 Weibull-Rayleigh Distribution (Ahmad et al., 2017) Corollary 5. If \=2 and _ = 12𝛽 , in Equation (36) the s th mo- ment of the rth order statistics of the WR distribution is given by ` s r:nWR =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) ks X(2𝛽)s/2 ( 1 m )( s2𝛼 +1) Γ ( s 2𝛼 + 1 ) (49) The mean of order statistics of WR distribution is derived and given by `r:nWR =  n (r−1)!(n−r)! ∑r−1 i=0 (−1) i ( r − 1 i ) k X(2𝛽)1/2 ( 1 m )( 12𝛼 +1) Γ ( 1 2𝛼 + 1 ) (50) The mean of the minimum X(1:n) order statistics of WR distri- bution is given by `1:nWR = nk(2𝛽)1/2Γ ( 1 2𝛼 + 1 ) (51) The mean of the maximum X(n:n) order statistics of WR dis- tribution is given by `n:nWR =  ∑n−1 i=0 (−1) i ( n − 1 i ) nk(2𝛽)1/2 X ( 1 i+1 )( 12𝛼 +1) Γ ( 1 2𝛼 + 1 ) (52) Corollary 6. The sth moment of the WR distribution de- veloped by Ahmad et al. (2017) is given by ` ′ s = (2𝛽) s/2ksΓ ( s 2𝛼 + 1 ) (53) Proof: If r=n=1 and \=2 and _ =1/2 𝛽 in Equation (36) the desired result is obtained The mean of a random variable X from the WR distribution is E(X) = (2𝛽)1/2kΓ ( 1 2𝛼 + 1 ) (54) Remark: Similar results can be deduced for the Rayleigh- Rayleigh distribution developed by Ateeq et al. (2019) and for other models presented in Table 1. 3.2 Parameter Estimation and Simulation This section is for determining the estimates of parameters of WEP distribution using the method of maximum likelihood estimation (MLE). A Simulation study is also conducted to assess the performance of the procedure. © 2023 The Authors. Page 273 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 3.2.1 Maximum Likelihood Estimation Let X1, X2,..., Xn,be independent and identically distributed random sample of size n from the WEP distribution with PDF derived as g(x) in Equation (9) with a set of parame- ters 𝜑=(𝛼,\,_ ,k). The likelihood function of the distribution is obtained as; Lik[g(x, 𝜑)] = n∏ i=1 [ n𝛼_ \ k ( x k ) (\−1) { _ ( x k )\ }(𝛼−1) exp ( − { _ ( x k )\ }𝛼 )] (55) The log-likelihood function logLik[g(x,𝜙)] denoted as LL is LL =  nlog𝛼 + nlog_ + nlog\ − nlogk+ (\ − 1) ∑ logx − n(\ − 1)logk +n(𝛼 − 1)log_ − n(𝛼 − 1)logk\ +(𝛼 − 1) ∑ logx\ − ∑ { _ ( x k )\ }𝛼 (56) The normal Equations are obtained as derivatives of LL for the parameters 0 = dLL d𝛼 =  n 𝛼 + nlog_ − nlogk\ + ∑ logx\ − ∑ { _ ( x k )\ }𝛼 log ( _ ( x k )\ ) (57) 0 = dLL d_ = n _ + n𝛼 _ + 𝛼 _ ∑︁ { _ ( x k )\ }𝛼 (58) 0 = dLL dk =  −nk − n(\−1) k + n(𝛼−1) k + 𝛼\k ∑ { _ ( x k )\ }𝛼 (59) 0 = dLL d\ =  n \ + ∑ logx − nlogk − n(𝛼 − 1) \k +(𝛼 − 1) ∑ \ x − 𝛼 ∑ { _ ( x k )\ }𝛼 log ( x k ) (60) A numerical solution to the above equations is adopted for the estimates of the parameters which are easier using the statistical software. 3.2.2 Simulation Study Assessment of the estimation of parameter procedure is per- formed by conducting a simulation study using the R-statistical software as follows; 1. Simulated data are generated using Equation (14) given by X = K { 1 _ [ −log(1 − u) 1 𝛼 ]} 1\ 2. The sample sizes taken are n = 20, 50, 250, 350, 500 3. Two sets I and II of parameter values are defined as I = (𝛼=0.5, \=1, _ =2.5, k=1.5) and II = (𝛼=1.0, \=2, _ =3.5, k=2.0) 4. Replicate the process for each sample size N=10,000 number of times 5. Compute the MSE by using MSE ∅ = 1N ∑N i=1 (∅i’- ∅) 2 where ∅’irepresents WEP parameters 6. Step five is carried our repeatedly for each parameter. The estimated values are obtained for the Bias, Mean Square Error (MSE), Root Mean Square Error (RMSE), and standard errors. The simulation study shows that the parameters of the distribution are stable in addition; the consistency of the MSE values for the maximum likelihood estimations implies that the estimation procedure is adequate. The results revealed that the MSE and RSME decrease as the sample size increases for both sets of actual values of the parameter. The standard errors also converge to zero as the sample size increases. The results from the simulation studies are presented in Table 2 for the first set of parameters and Table 3 for the second set of parameters. 3.3 Application to Lifetime Datasets The usefulness of the distribution is demonstrated by analyzing three real-life datasets using the R software (Core Team, 2013) . The best-fitted model is usually identified with the smallest values of goodness-of-fit criteria which are the Log-likelihood (LL), Akaike information criterion (AIC), Bayesian informa- tion criterion (BIC), Consistent Akaike information criterion (CAIC), Hannan-Quinn information criterion (HQIC), the Kolmogorov statistics and P-value. The criteria are defined as follows; AIC=-2LL+2c; BIC=-2LL+clog(n); CAIC=-2LL+ 2c(n−2)n−c−2 ; HQIC = 2log (log(n)(c − 2LL)). The performance of the WEP distribution is compared with some similar families of distribu- tions and some notable models existing in the literature with the density functions defined by 1. Generalized Exponential Weibull (GEW) distribution GEW (x; 𝛼, 𝛽 , \ , k) = \(𝛼 + 𝛽kx(k−1)) exp(−(𝛼x + 𝛽xk))(1 − exp(𝛼x + 𝛽xk))\−1 2. Transmuted New Weibull Pareto (TNWP) distribution TNWP(X;_ , 𝛽 , \ , k) = 𝛽 \ k ( x k ) ( 𝛽 −1) e(−\( x k ) 𝛽 ( 1 − _ + 2_e−\( x k ) 𝛽 ) © 2023 The Authors. Page 274 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 3. Kumaraswamy Exponential Pareto (KEP) distribution KEP(X; 𝛼, 𝛽 , \ , k) = 𝛼 𝛽_ \ k ( x k ) (\−1) e−_ ( x k ) \ ( 1 − e−_ ( x k ) \ )𝛼−1 ( 1 − ( 1 − e−_ ( x k ) \ )𝛼 ) 𝛽 −1 4. Gompertz Exponential Pareto (GEP) distribution GEP(X; 𝛼, _ , 𝛽 , \ , k) = 𝛼_ \ k ( x k ) (\−1) ( e−_ ( x k ) \ )− 𝛽 e 𝛼 𝛽 [ 1− ( e −_ ( xk ) \ )− 𝛽 ] 5. Beta Exponential Pareto (BEP) distribution BEP(X; 𝛼, _ , 𝛽 , \ , k) = 𝛼\ _B(𝛼 𝛽) ( x _ ) (\−1) e−𝛼 𝛽 ( x _ ) \ [ 1 − e− 𝛽 ( x _ ) \ ]𝛼−1 3.3.1 Application to the Hydrological Data Set The first data contain 72 exceedances of flood peaks (inm3/s) of the Wheaton River discharge near Carcross in Yukon Territory, Canada for the years 1958-1984. The data set has been applied by several authors including Aryal (2019) using BEP, Tahir and Akhter (2018) using TNWP, and recently by Adeyemi et al. (2021) for evaluating the performance of the GEP distribution. Figure 3. Plots of PDFs and CDFs of Fitted Distributions Fitted to the Hydrological Data First Data Set: 1.7, 2.2, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.4, 18.7, 8.5, 25.5, 11.6, 14.1, 22.1, 1.1, 0.6, 2.2, 39.0, 0.3, 15.0, 11.0, 7.3, 22.9, 0.9, 1.7, 7.0, 20.1, 0.4, 2.8, 14.1, 9.9, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4, 11.9, 21.5, 1.5, 2.5, 27.4, 1.0, 27.1, 20.2, 16.8, 5.3, 1.9, 10.4, 13.0, 10.7, 12.0, 30.0, 9.3, 3.6, 2.5, 27.6, 14.4, 36.4, 1.7, 2.7, 37.6, 64.0, 1.7, 9.7, 0.1, 27.5, 1.1, 2.5, 0.6, 27.0. The values of the estimated parameters of the competing models and the goodness-of-fit criteria are displayed in Table 4 and Table 5 respectively. The visual result of goodness-of-fit from data analysis in form of the histogram with the estimated densities and the CDFs of the fitted models are displayed in Figure 3. Table 4 revealed that the WEP distribution com- petes favorably with the GEP model and performs better than TNWP, KEP, and BEP distributions based on the smallest values of goodness-of-fit statistics. Figure 3 supported the se- lection of the proposed model as more flexible than the other models for the dataset. 3.3.2 Application to Tensile Strength of Polyester Fibers The second data represents 30 measurements of the tensile strength of polyester fibers available in (Quesenberry and Hales, 1980) . Figure 4. Plots of Histogram with Densities and CDFs of Fitted Distributions to the Tensile Strength Data Second Data Set: 0.023, 0.032, 0.054, 0.069, 0.081, 0.094, 0.105, 0.127, 0.148, 0.169, 0.188,0.216, 0.255, 0.277, 0.311, 0.361, 0.376, 0.395, 0.432, 0.463, 0.481, 0.519,0.529, 0.567, 0.642, 0.674, 0.752, 0.823, 0.887, 0.926 The computed results of estimated parameters and the goodness-of-fit criteria are displayed in Table 6 and Table 7 respectively for the models under consideration. Figure 4 is the pictorial display of goodness-of-fit from data analysis in form of the histogram with the estimated densities and CDFs of the fitted models to the tensile strength data set. The fit of the WEP model is compared with TNWP, GEP, GEW, and KEP models. Results displayed in Table 6 revealed that the WEP model has more flexible capabilities for the data than the competitors. The visual results from the estimated densities and CDFs in Figure 4 strengthened the choice of the WEP model. 3.3.3 Application to Lifetimes of Steel Specimens The data represents the lifetimes (t) of steel specimens tested at stress (s) level of s=38.5 obtained from the 14 different stress levels reported in (Lawless, 2011) . Third Data Set: 60, 51, 83, 140, 109, 106, 119, 76, 68, 67, 111, 57, 69, 75, 122, 128, 95, 87, 82, 132. The lifetime data is fitted to the WEP model and compared with GEP, GEW, and KEP models. Computed results of estimated parameters and the goodness-of-fit criteria are presented in Table 8 and Table 9 respectively. The plots for the densities and CDFs are displayed in Figure 5. Application of the proposed models to the data set © 2023 The Authors. Page 275 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 Figure 5. Plots of PDFs and CDFs of Fitted Distributions to the Steel Specimens Data of steel specimens revealed the suitability of the WEP model as a better lifetime distribution for fitting the data compared to some other existing distributions in the literature. The WEP model provides the smallest AIC, CAIC, and BIC and the plots in Figure 5 substantiate our choice of WEP distribution. 4. CONCLUSION The WEP model is a new lifetime distribution with adequate potential for analyzing left-skewed, right-skewed, and approx- imately symmetric phenomena from the field of hydrology, reliability engineering, actuarial and finance, and public health. It is also suitable for fitting real-life datasets in other areas of ap- plications characterized by risky kurtosis. It is confirmed in this study that WEP has superior performance than the competing distributions for modeling the reliability and the hydrological data sets and the lifetime of steel specimens. The discover- ies from the theory of order statistics extended in the study established a novel area of study in distribution theory which also contributed to the conclusion that the WEP distribution provides sufficient characterizations for WR, EP, RR distribu- tions, and some other existing lifetime distributions. Other important areas of further research include the estimation of parameters by order statistics and the real-life application of moments of order statistics in predictive modeling. 5. ACKNOWLEDGMENT The authors are grateful for the contributions and suggestions of the Editors, the Editor-in-Chief and all the anonymous re- viewers which improve the original manuscript. Special thanks to Mr. John Anih for the payment of the APC on behalf of the corresponding Author. REFERENCES Adeyemi, A. O., E. E. Akarawak, and I. A. Adeleke (2021). The Gompertz Exponential Pareto Distribution with the Properties and Applications to Bladder Cancer and Hydro- logical Datasets. Communications in Science and Technology, 6(2); 107–116 Ahmad, A., S. Ahmad, and A. Ahmed (2017). Characteriza- tion and Estimation of Weibull-rayleigh Distribution with Applications to Life Time Data. Applied Mathematics and Information Sciences Letters, 5; 71–79 Akarawak, E. E., I. A. Adeleke, and R. O. Okafor (2013). The Weibull-Rayleigh Distribution and Its Properties. Journal of Engineering Research, 18(1); 56–67 Akarawak, E. E., I. A. Adeleke, and R. O. Okafor (2017). 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Simulation Study for Set of Parameters I=(𝛼=0.5,\=1,_ =2.5,k=1.5) Bias MSE n â _̂ \̂ k̂ â _̂ \̂ k̂ 20 0.6987 -1.2998 0.1999 -0.2995 0.8459 2.0494 0.3998 0.4515 50 0.6990 -1.3005 0.2006 -0.3007 0.6336 1.8353 0.1832 0.2352 250 0.6987 -1.3006 0.2003 -0.2991 0.5183 1.7200 0.0691 0.1185 500 0.7000 -1.3001 0.2001 -0.2989 0.5044 1.7048 0.0544 0.1042 1000 0.6986 -1.3000 0.1998 -0.2992 0.4973 1.6972 0.0471 0.0972 RMSE St.Error n â _̂ \̂ k̂ â _̂ \̂ k̂ 20 0.9198 1.4316 0.6323 0.6719 0.5981 0.5999 0.5998 0.6015 50 0.7960 1.3547 0.4208 0.4849 0.3793 0.3795 0.3781 0.3804 250 0.7199 1.3115 0.2628 0.3442 0.1695 0.1688 0.1701 0.1704 500 0.7102 1.3057 0.2334 0.3228 0.1199 0.1202 0.1199 0.1196 1000 0.7052 1.3028 0.2171 0.3117 0.0848 0.0846 0.0848 0.0851 Table 3. Simulation Study for Set of Parameters II=(𝛼=1.0,\=2,_ = 3.5,k=2.0) Bias MSE n â _̂ \̂ k̂ â _̂ \̂ k̂ 20 -0.0523 -2.5528 -1.0522 -1.0528 0.0150 6.5292 1.1193 1.1208 50 -0.0526 -2.5526 -1.0525 -1.0527 0.0077 6.5207 1.1126 1.1130 250 -0.0526 -2.5526 -1.0527 -1.0525 0.0038 6.5168 1.1092 1.1088 500 -0.0526 -2.5526 -1.0526 -1.0524 0.0032 6.5161 1.1086 1.1082 1000 -0.0526 -2.5526 -1.0525 -1.0523 0.0030 6.5160 1.1080 1.1081 RMSE St.Error n â _̂ \̂ k̂ â _̂ \̂ k̂ 20 0.1225 2.5552 1.0579 1.0587 0.1108 0.1109 0.1107 0.1111 50 0.0875 2.5535 1.0548 1.0549 0.0699 0.0701 0.0701 0.0702 250 0.0612 2.5528 1.0532 1.0530 0.0314 0.0314 0.0313 0.0313 500 0.0571 2.5526 1.0528 1.0526 0.0221 0.0222 0.0221 0.0220 1000 0.0549 2.5526 1.0526 1.0526 0.0156 0.0157 0.0156 0.0156 Table 4. Log-likelihood and Maximum Likelihood Estimates of the Parameters for the Flood Peaks Data Model â 𝛽 _̂ \ k̂ -LL GEP 0.6735 0.2594 0.3099 0.7474 0.8814 499.7330 WEP 0.0786 - 3.1539 11.1779 10.1814 501.1056 TNWP 0.4514 - 0.4092 6.7312 0.8771 502.9130 KEP 0.9245 0.2594 0.3099 0.7624 0.8814 501.1506 BEP 0.0332 0.4984 0.7474 0.2985 0.5482 501.9790 Table 5. Goodness-of-fit Statistics for the Flood Peaks Data Model AIC CAIC BIC HQIC K-S p-value WEP 509.1056 509.7026 518.2122 512.7310 0.1067 0.3850 GEP 509.7330 510.6461 521.1164 514.2648 0.1029 0.4310 TNWP 510.9130 511.5100 520.0197 514.5384 0.1069 0.3812 KEP 511.1506 512.0589 522.5340 515.6824 0.1071 0.3802 BEP 511.9590 512,8680 523.3420 516.4910 - - © 2023 The Authors. Page 277 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 Table 6. Log-likelihood and Estimated Parameters for the Tensile Strength Data Model â 𝛽 _̂ \ k̂ -LL WEP 0.0402 - 2.4076 32.9847 0.3891 -1.7503 TNWP 6.5409 - 0.2768 1.7556 1.3982 -1.7377 GEP 15.9217 30.1267 0.3245 0.9161 4.9943 -2.3263 GEW 3.1808 - 2.9046 21.1988 0.1269 -1.2846 KEP 0.1795 13.5155 11.7830 7.4832 4.1216 -2.2521 Table 7. Goodness-of-fit Statistics for the Tensile Strength Data Model AIC CAIC BIC HQIC K-S p-value WEP 4.4993 6.0993 10.1041 6.2923 0.0826 0.9760 TNWP 4.5246 6.1246 10.1294 6.3176 0.0853 0.9679 GEP 5.3473 7.8473 12.3533 7.5886 0.0859 0.9661 GEW 5.4308 7.0308 11.0357 7.2238 0.0884 0.9568 KEP 5.4959 7.9959 12.5018 7.7371 0.0917 0.9427 Table 8. Log-likelihood and Estimated Parameters for the Steel Specimen Data Model â 𝛽 _̂ \ k̂ -LL WEP 1.9542 - 33.5981 1.409 9.6908 102.9148 GEP 0.0065 0.7775 0.5749 5.6195 1.1537 102.9122 GEW 0.0232 - 1.0449 10.9496 -0.1876 104.4920 KEP 9.1038 3.5236 0.1109 0.6503 1.4219 105.6481 Table 9. Goodness-of-fit Statistics for the Steel Specimen Data Model AIC CAIC BIC HQIC K-S p-value WEP 213.8297 216.4964 217.8126 214.6072 0.1057 0.9615 GEP 214.8244 219.8031 219.8031 215.7963 0.1096 0.9485 GEW 214.8244 219.6506 220.9669 217.7615 0.1387 0.7871 KEP 221.2962 225.5819 226.2748 222.2680 0.1405 0.7744 © 2023 The Authors. Page 278 of 279 Adeyemi et. al. Science and Technology Indonesia, 8 (2023) 265-279 Pareto Distribution. Far East Journal of Theoretical Statistics, 50(1); 31–49 Nasiru, S. and A. Luguterah (2015). The New Weibull-Pareto Distribution. Pakistan Journal of Statistics and Operation Re- search, 11(1); 103–114 Quesenberry, C. P. and C. Hales (1980). Concentration Bands for Uniformity Plots. Journal of Statistical Computation and Simulation, 11(1); 41–53 Rashwan, N. I. and M. M. Kamel (2020). The Beta Exponential Pareto Distribution. Far East Journal of Theoretical Statistics, 58; 91–113 Rayleigh, L. (1896). The Theory of Sound, volume 2. Macmillan Rényi, A. (1961). 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Page 279 of 279 INTRODUCTION EXPERIMENTAL SECTION Materials and Method The New Extension of Exponential Pareto Distribution Sub-Models of WEP Distribution Properties of the Weibull Exponential Pareto Distribution The Reliability and the Hazard Rate Function Asymptotic Behavior of WEP Quantile Function, Simulation, and Median Moments of WEP Distribution The Mean, Variance, Skewness, and Kurtosis of WEP Distribution Moment Generating Function of WEP Distribution Relationship between WEP and the Weibull-Pareto (WPD) Distribution Shannon Entropy of Weibull Exponential Pareto Distribution Renyi Entropy of Weibull Exponential Pareto Distribution Distribution of the Order Statistics of WEP Distribution Moments, Mean, and Variance of Order Statistics of WEP Distribution RESULTS AND DISCUSSION Generalization of Properties of Order Statistics for Some Lifetime Distribution Exponential Pareto Distribution 5 Weibull-Rayleigh Distribution 2 Parameter Estimation and Simulation Maximum Likelihood Estimation Simulation Study Application to Lifetime Datasets Application to the Hydrological Data Set Application to Tensile Strength of Polyester Fibers Application to Lifetimes of Steel Specimens CONCLUSION ACKNOWLEDGMENT