EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3856-3898 ISSN 1307-5543 – ejpam.com Published by New York Business Global Analyzing Cancer Survival Times Using the Exponentiated Fréchet-Gompertz Distribution Ibtesam Ali Alsaggaf 1 Department of statistic, Faculty Science, King Abdul-Aziz University, Jeddah, 21589, Saudi Arabia Abstract. Understanding the survival of cancer patients is essential for determining optimal treat- ment strategies. This research introduces a robust distribution for analyzing survival data known as the Exponential Fréchet-Gompertz distribution (EFG). The EFG combines the unique characteris- tics of the Exponential Fréchet and Gompertz distributions, enhancing its efficiency and providing greater flexibility in representing complex datasets. The study specifically investigates the EFG distribution’s effectiveness in modeling cancer patients’ survival times. A comprehensive analysis of the distribution’s properties is presented, including the quantile and quartile functions, shape indices, moments, moment-generating function, characteristic function, mean residual life, mean waiting time, Rényi entropy, and order statistics. The parameters of the distribution are derived using five distinct methods: Maximum Likelihood Estimation (MLE), Ordinary Least Squares (OLS), Weighted Least Squares (WLS), Cramér-von Mises (CVM), and Maximum Product of Spacings (MPS). A Monte Carlo simulation technique is employed to evaluate the performance of these estimation methods. The simulation results indicate that as sample size increases, the mean square error (MSE) values for all estimators decrease. Notably, the MLE exhibits the lowest MSE, while the MPS has the highest MSE, particularly for smaller sample sizes. Furthermore, the study presents a comprehensive comparison of the effectiveness of these estimation methods in analyzing survival times for various cancer types, including bladder, bone, blood and brain cancer. The results indicate that the EFG distribution is an optimal model for representing the survival times of patients across these different cancers data. Furthermore, the W.LS method yielded superior estimations for most datasets concerning cancer patient survival. Overall, the EFG distribution demonstrates exceptional capability in accurately fitting survival times for cancer patients com- pared to other competing distributions. 2020 Mathematics Subject Classifications: 62E10 , 62N05 Key Words and Phrases: Estimation method, probability distributions, maximum likelihood method, characteristic function, Exponentiated Fréchet distribution, Gompertz distribution DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5471 Email address: ialsaggaf@kau.edu.sa (I.A. Alsaggaf) https://www.ejpam.com 3856 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3857 1. Introduction Estimating the survival time for cancer patients is a crucial aspect of cancer treatment. It helps doctors and patients make informed decisions about the treatment plan and manage expectations regarding the outcome. Researchers are working diligently to develop accurate models and statistical estimation techniques that can be used to predict the lifetime of cancer patients. [28] analyzed the lifetime of patients suffering from leukemia using a generalized linear exponential distribution. [34] employed McDonald log-logistic distribution with maximum likelihood technique to estimate the lifetime for breast cancer patients. [9] proposed the Alpha Power Weibull–Exponential model and used it to estimate the survival time for head and neck cancer patients. [24] conducted a survival analysis of cancer patients using Alpha power Kumaraswamy Weibull distribution with maximum likelihood estimation technique. Researchers have put significant effort into developing various methods for creating new distributions that can adapt to represent different types of data, particularly survival data effectively. Techniques such as compounding, adding parameters, composing, and transforming have been advanced to broaden the scope of distributions. In 2013, [3] pro- posed a general approach known as the T-X transformation, that enables the use of any baseline distribution to generate a new distribution. The Lomax-G family by [14], the Weibull-G family by [10], the Lindley-G family by [11], the power Lindley-G family by [19], the Gompertz-G family by [2] are some generated families of distributions by T-X transformation technique. [21] introduced the new lifetime exponential-X family which used to generate a variety of distributions, such as the exponential Fréchet distribution [4], the exponentiated Weibull [5], the exponential inverted Topp-Leone distribution [30]. Furthermore, [8] provided the Exponentiated Fréchet generator of distribution. This fam- ily seems to be a great fit for modeling complex data and is particularly useful in reliability analysis. The cumulative function (CDF) and the probability density (PDF) of this family is given as F (x) = 1 − [ 1 − exp { − ( λ −log[1 − G(x; Θ)]) )β }]α , x > 0; λ, γ, θ, β, α > 0. (1) f(x) = αβλβ g(x; Θ) 1 − G(x; Θ) {−log[1 − G(x; Θ)]}−(β+1) exp { − ( λ −log[1 − G(x; Θ)] )β } (2) where Θ represents the parameter vector of the baseline distribution G. The Gompertz distribution is a statistical distribution utilized in survival analysis to model lifetime data. This traditional distribution is employed to model the survival func- tion based on mortality laws and is crucial in estimating various life-related events, such as human mortality rates and financial outcomes. This makes it a suitable choice for repre- senting survival time for cancer patients. The Gompertz distribution was first introduced I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3858 by [18]. The CDF of Gompertz is F (x) = 1 − e − λ β (eβx−1), x > 0; λ, β > 0. (3) and its PDF is f(x) = λeβxe − λ β (eβx−1) (4) While this distribution effectively models the survival function, its capacity to represent various lifetime data is restricted due to its tendency to display an exponentially increasing failure rate over lifetime data. This characteristic makes it unsuitable for certain appli- cations. As a result, scientists developed the Gompertz distribution and explored various generalizations and combinations with other distributions to enhance its performance. Beyond the development of the Gompertz distribution, [2] introduced the Gompertz gen- eralized family utilizing the T − X transformation technique. This approach led to the creation of several derived distributions, including the Gompertz Normal, Gompertz Beta, Gompertz Gamma, Gompertz Log-Logistic, Gompertz Exponentiated Weibull, and Gom- pertz Lomax [32]. [23] implemented the same transformation to create a new model called the generalized Gompertz-G family. [16] introduced a generalized Gompertz distribution with three parameters utilizing the exponentiated method. Building on this foundation, [15] enhanced [16] model by incorporating two additional shape parameters through the exponentiated generalized technique proposed by [13]. Furthermore, [1] combined the ex- ponentiated approach in [16] with a Gompertz exponential derived from [2], resulting in a new generalized Gompertz distribution. This model offers greater flexibility for analyzing survival data. Another significant advancement is the Kumaraswamy-G generalized Gom- pertz distribution, proposed by [17]. Recently, [7] introduced a new family of distributions by combining the odd Weibull family with the inverse Gompertz distribution, merging features from both. This research aims to develop the exponential Frechet-Gompertz (EFG) distribution by amalgamating the distinctive characteristics of both the exponential Frechet and Gom- pertz distributions into a single model. The research entails an exhaustive examination of the distribution’s characteristics, including the estimation of distribution parameters through the utilization of five distinct estimation methods. Moreover, the study includes a meticulous comparison of the efficacy of these estimation methods in analyzing the sur- vival time for several types of cancer. This article is classified as follows: Section 2 describes the EFG using graphical rep- resentations. Section 3 gives useful expansion for the EFG’s CDF and PDF. Section 4 derived statistical properties for EFG. In section 5, five estimation methods are employed to estimate the EFG parameters: maximum likelihood (MLH), ordinary least squares (O.LS), weighted least squares (W.LS), Cramér-von Mises (CRM) and maximum product of spacing (MPS). Section 6 presents assessing the performance of the estimation methods using Monte Carlo simulation studies . Section 7 investigates different datasets of cancer I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3859 patient data are analyzed to assess the EFG’s modeling effectiveness and compare its per- formance against competing distributions. Section 8 concludes with some final remarks. 2. Exponentiated Fréchet Gompertz distribution The EFG’s CDF and PDF are found by replacing the G(x) and g(x) in (1) and (2) by (3) and (4) as follows: F (x) = 1 − [ 1 − exp { − ( λ γ(exθ − 1) )β }]α , x > 0; λ, γ, θ, β, α > 0. (5) f(x) = αβθλβexθ γβ exp { − ( λ γ(exθ−1) )β } [(exθ − 1)](β+1) [ 1 − exp { − ( λ γ(exθ − 1) )β }]α−1 (6) The survival and the hazard rate functions of EFG are given respectively as follows S(x) = [ 1 − exp { − ( λ γ(exθ − 1) )β }]α (7) H(x) = αβθλβexθ γβ exp { − ( λ γ(exθ−1) )β } [(exθ − 1)](β+1) [ 1 − exp { − ( λ γ(exθ − 1) )β }]−1 (8) (a) f(x) (b) H(x) Figure 1: The density and hazard function plots I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3860 Figure 1a shows a variety of shapes of EFG distribution, symmetric and asymmetric with different degrees of skewness. Additionally, Figure 1b represents the hazard function of EFG which exhibits a range of behaviors that demonstrate its capacity to adapt to different data types, indicating a high level of flexibility. 3. Useful Expansion for the EFG’s CDF and PDF This subsection provides the expansion for the EFG’s Cumulative and Probability Functions. The binomial series (9), (10) and the exponential function (11) given below are applied to expand the EFG’s CDF and PDF in (5) and (6), respectively. (1 − r)n = ∞∑ l=0 (−1)l ( n l ) rl, |r| < 1, n is any real number (9) (1 − u)−n = ∞∑ w=0 ( n + w − 1 w ) uw. (10) e−z = ∞∑ v=0 (−1)v(z)v v! , (11) 3.1. CDF Expansion First, the binomial series (9) is applied to expand (5). The EFG’s CDF might then be written as F (x) = 1∑ l1=0 ∞∑ l2=0 (−1)l1+l2 ( 1 l1 )( αl1 l2 ) exp { − ( λ γ (eθx − 1) )βl2 } Then, the exponential function (11)is utilized resulting the following form for F (x): F (x) = 1∑ l1=0 ∞∑ l2=0 ∞∑ v1=0 (−1)l1+l2+v1−l2v1β v1! ( 1 l1 )( αl1 l2 )( λ γ )l2v1β ( 1 − eθx )−l2v1β By applying the series in (10), the EFG’s CDF can be reduced to F (x) = η1eθxw1 (12) where I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3861 η1 = 1∑ l1=0 ∞∑ l2=0 ∞∑ v1=0 ∞∑ w1=0 (−1)l1+l2+v1−l2v1β v1! ( 1 l1 )( αl1 l2 )( l2v1β + w1 − 1 v1 )( λ γ )l2v1β (13) 3.2. PDF Expansion To find the expansion for the EFG’S PDF, first, the series (9) and (11) are employed to expand (6). Then the f(x) might be written as below. f(x) = ∞∑ l3=0 ∞∑ v2=0 ∞∑ v3=0 (−1)l3+v2+v3−β(l3v2+v3+1)−1 v2!v3! ( α − 1 l2 )( λ γ )β(l3v2+v3+1) αβθeθx (1 − eθx)β(l3v2+v3+1)+1 Follow that, the series (10) is applied to reduce the EFG’s PDF to the following form. f(x) = η2 α β θ eθ(1+w2)x, (14) where η2 = ∞∑ l3=0 ∞∑ v2=0 ∞∑ v3=0 ∞∑ w2=0 (−1)l3+v2+v3−β(l3v2+v3+1)−1 v2!v3! ( α − 1 l2 )( β(l3v2 + v3 + 1) − 1 w3 ) (15) 4. Statistical Properties of the EFG This section derives several statistical properties of the EFG distribution, including the quantile function, moments, moment-generating function, characteristic function, Rényi entropy, and order statistics. Statistical properties are vital for comprehensively describing and analyzing data from diverse perspectives. 4.1. Quantile Function and Quartiles The quantile function of EFG is written as Qp = ln 1 − βλ γ 1 ln [ 1 − (1 − p) 1 α ]  /β, 0 < p < 1. (16) The median of the EFG distribution can be obtained as Q(0.5) = ln 1 − βλ γ 1 ln [ 1 − (1 − 0.5) 1 α ]  /β, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3862 Hence, the 25th percentile and the 75th percentile of the EFG distribution are given as Q(0.25) = ln 1 − βλ γ 1 ln [ 1 − (1 − 0.75) 1 α ]  /β, Q(0.75) = ln 1 − βλ γ 1 ln [ 1 − (1 − 0.25) 1 α ]  /β, 4.2. Shape Indices The shape of the EFG can be assessed using Galton’s skewness and Moors’ kurtosis [31], which can be calculated by utilizing the quantile function (16) and respectively given as follows: Skewness = Q0.75 − 2Q0.5 + Q0.25 Q0.75 − Q0.25 , and Kurtosis = Q0.875 − Q0.625 + Q0.375 − Q0.125 Q0.75 − Q0.25 . 4.3. Moments The moment of of X for f(x) is written as E (xr) = ∫ ∞ 0 xrf(x)dx If X follows the EFG (λ, γ, θ, β, α), then the rth moment of X is written as E (xr) = η2 α β θ ∫ ∞ 0 xreθ(1+w2)xdx, where η2 is given by (15). By using Laplace transformation, Lt [f(t)] (s) = ∫∞ 0 f(t)e−stdt where f(t) is defined for t ≥ 0 [25], with taking f(t) = tr then ∫∞ 0 f(t)e−stdt = r! sr+1 . Therefore, f(x) = xr then ∫∞ 0 f(x)e−sxdx = r! sr+1 , where s = −θ(1 + w2). Then the rth moment is expressed as µr = E(xr) = η2 α β θ ( −1 θ(1 + w2) )r+1 Γ(r + 1), r ≥ 0 (17) Then, the EFG’s mean is written as µ = E (x) = η2 α β θ ( 1 1 + w2 )2 (18) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3863 The EFG’s variance is determined by σ2 =E(x2) − µ2 = 2 η2 α β θ2 ( −1 1 + w2 )3 − µ2, where η2 is given by (15). 4.4. Moment Generating Function The moment generating function (MGF) of X for f(x) is written as Mx(t) = E(etx) = ∫ ∞ 0 etxf(x)dx If X follows the EFG (λ, γ, θ, β, α), then the EFG’s MGF is written as Mx(t) = E(etx) = η2 α β θ ∫ ∞ 0 etx eθ(1+w2)xdx, By using Laplace transformation, with taking f(t) = 1 then ∫∞ 0 f(t)e−stdt = 1 s . Therefore, f(x) = 1 then ∫∞ 0 f(x)e−sxdx = 1 s , where s = −(t + θ(1 + w2)). Then the then MGF will be given as Mx(t) = E(etx) = η2 α β θ ( −1 t + θ(1 + w2) ) , (19) where η2 is given by (15). 4.5. Characteristic Function The characteristic function of EFG is simply constructed as : ϕx (t) =E(eitx) = η2 α β θ ( −1 it + θ(1 + w2) ) , (20) where η2 is given by (15). 4.6. Mean residual life and mean waiting time If X ∼ EFG(λ, γ, θ, β, α) with S(t) provided in (7) , then the mean residual life, µ(t), is expressed as µ(t) = 1 S(t) ( E(t) − ∫ t 0 xf(x)dx ) − t. (21) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3864 If the incomplete moment, Iinc = ∫ t 0 xf(x)dx, then Iinc = η2 α β θ ∫ t 0 xeθ(1+w2)xdx. Setting y = θ(1 + x2)x, then simplifying will be obtained, Iinc = ( η2αβ θ(1 + w2)2 )∫ θ(1+w2)t 0 yeydy, Using integration by part by taking u = y and dv = eydy, the incomplete moment will be obtained as Iinc = ( η2αβ θ(1 + w2)2 ) [ θ(1 + w2)teθ(1+w2)t − eθ(1+w2)t + 1 ] , (22) Substituting (18), and (22) in (21), µ(t) might be rewritten as µ(t) = 1 S(t) η2αβ θ(1 + w2)2 [ eθ(1+w2)t − θ(1 + w2)teθ(1+w2)t ] − t, In a similar manner, the mean waiting time, µ̄(t), might be defined as µ̄(t) = t − 1 F (t) ∫ t 0 xf(x)dx, (23) where F (t) is provided in (12). Then, µ̄(t) of the EFG can be found by substituting (12) and (22) in (23) as follows µ̄(t) = t − ( η2αβ η1θ(1 + w2)2eθxw1 ) [ θ(1 + w2)teθ(1+w2)t − eθ(1+w2)t + 1 ] Mean residual life and mean waiting time are important concepts in reliability the- ory and queuing theory, respectively, as they provide insights into the expected remain- ing lifespan of a system or the time until an event occurs. Incomplete moments, which summarize essential features of distributions without requiring full data (censored data), complement these concepts by offering partial insights into uncertainty and variability. They are particularly useful in situations where data is limited or when focusing on es- sential features is necessary, such as in risk assessment and model simplification. Derived results from incomplete moments offer partial insights that can inform decision-making and enable comparative analyses, despite acknowledging the inherent uncertainty in not having a full distribution. Together, Mean residual life and mean waiting time can guide decision-making in fields like finance and operations, while the interpretations of their de- rived results enable comparisons and assumptions about system performance, even when data is limited. I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3865 4.7. Rényi entropy REX(ζ) is the Rényi entropy function which given as REX(ζ) = 1 1 − ζ log (∫ ∞ 0 f(x)ζdx ) ; ζ > 0, ζ ̸= 1. Then, applying the EFG’s PDF in (6) f(x)ζ = ( αβθλβ γβ )ζ eθxζ exp { − ( λ γ(exθ−1) )βζ } [(exθ − 1)](β+1)ζ [ 1 − exp { − ( λ γ(exθ − 1) )β }](α−1)ζ Applying the same approach in Subsection 3 and using (9), (10) and (11), then f(x)ζ = η∗ 2 αζ βζ θζ eθ(ζ+w3)x, where η∗ 2 = ∞∑ l4=0 ∞∑ v4=0 ∞∑ v5=0 ∞∑ w3=0 (−1)l4+v4+v5−β(l4v4+v5ζ+ζ)−ζ v4!v5! ( (α − 1)ζ l4 )( β(l4v4 + v5ζ + ζ) − ζ + w3 − 1 w3 ) Using Laplace transformation, since f(t) = 1 then Lt [f(t)] (s) = 1 s , where s = −(ζ + w3)β , then the Rényi entropy of the EFG, is then will be reduced to REx(ζ) = 1 1 − ζ log [ −η∗ 2 αζ βζ θζ−1 (ζ + w3) ] . 4.8. Order statistics The density function, fj:n(x), of the jth order statistics is given as fj:n(x) = 1 B(j, n − j + 1)f(x) [F (x)]j−1 [1 − F (x)]n−j . By employing the series formula , fj:n(x) can be expressed as fj:n(x) = 1 B(j, n − j + 1) n−j∑ l5=0 (−1)l5 ( n − j l5 ) f(x) [F (x)]l5+j−1 . (24) By substituting the CDF (12) and PDF of EFG (14) into (24), the PDF of Xj:n is fj:n(x) = 1 B(j, n − j + 1) n−j∑ l5=0 (−1)l5 ( n − j l5 )[ η2 α β θ eθ(1+w2)x ] [ η1eθxw1 ]l5+j−1 (25) where η1 and η2 are given by (13) and (15), respectively. I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3866 5. Estimation methods 5.1. Maximam Likelihood Method (MLH) For a random sample x1, x2, ..., xn from EFG, the log-likelihood function (ℓ), for Θ = (λ, γ, θ, β, α), is written as L(Θ) = n ln α + n ln β + n ln θ + n β ln λ − n β ln γ + β n∑ i=1 xi − ( λ γ )β n∑ i=1 (eθxi − 1)−β − (β + 1) n∑ i=1 ln (eθxi − 1) + (α − 1) n∑ i=1 ln [ 1 − exp { − ( λ γ(eθxi − 1) )β }] (26) The following equations from (27) to (31) represent the partial derivation from ℓ func- tion, (26), regards to the parameters (λ, γ, θ, β, α). The ML estimates, λ̂MLH , γ̂MLH , θ̂MLH , β̂MLH and α̂MLH can be obtained by maximizing the equation (26) or by solving the equations from (27) to (31) using numerical iterative technique. ∂ℓ ∂λ = (α − 1) β · n∑ i=1  ( λ (eθxi −1)γ )β e − ( λ (eθxi −1)γ )β λ · 1 − e − ( λ (eθx−1)γ )β   − β λ · ( λ γ )β n∑ i=1 ( eθxi − 1 )−β + βn λ (27) ∂ℓ ∂γ = −(α − 1) β γ · n∑ i=1  ( λ (eθxi −1)γ )β e − ( λ (eθxi −1)γ )β 1 − e − ( λ (eθxi −1)γ )β   +βγ· ( λ γ )β n∑ i=1 ( eθxi − 1 )−β −βn γ (28) ∂ℓ ∂θ = n θ + − (β + 1) · n∑ i=1 { xiexiθ exiθ − 1 } + β · ( λ γ )β n∑ i=1 xieθxi ( eθxi − 1 )−β−1 I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3867 − ( (α − 1) · β · λβ γβ ) · n∑ i=1  xi · ( eθxi − 1 )−(β+1) e θxi− ( λ γ·(eθxi −1) )β 1 − e − ( λ γ·(eθxi −1) )β   (29) ∂ℓ ∂β = n β +n ln λ−n lnγ+ n∑ i=1 xi− n∑ i=1 ln(eθxi−1)− n∑ i=1 {( λ γ · (eθxi − 1) )β · ln ( λ γ (eθxi − 1) )} + (α − 1) · n∑ i=1  ( λ (eθxi −1)γ )β ln ( λ (eθxi −1)γ ) e ( λ (eθxi −1)γ )β − 1  (30) ∂ℓ ∂α = n α + n∑ i=1 ln 1 − e − ( λ (eθx−1)γ )β  (31) 5.2. Ordinary Least Square Method (O.LS) O.LS method is proposed by [33], which is based on the difference between the empirical and theoretical cdf. Suppose a random sample from EFG distribution with size n and X(1), X(2), ..., X(n) are its order statistics. The sum of squares for the difference between the empirical and theoretical cdf of EFG distribution is formulated in equation (32). U1(Θ) = n∑ i=1 1 − 1 − exp − ( λ γ(eθx(i) − 1) )β  α − ( i n + 1 )2 (32) where ( i n+1 ) is the empirical cdf and i = 1, 2, ..., n. The partial derivation from the equation (32) in regard to Θ = (λ, γ, θ, β, α) can be written as follows: ∂ℓ ∂λ = −2αβλ(β−1) γβ · n∑ i=1  1( eθx(i) − 1 ) β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3868 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  · e −  λ( e θx(i) −1 ) γ β (33) ∂ℓ ∂γ = 2αβλβ γ(β+1) · n∑ i=1  1( eθx(i) − 1 ) β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  · e −  λ( e θx(i) −1 ) γ β (34) ∂ℓ ∂θ = 2αβ ( λ γ )β · n∑ i=1 x(i) · ( 1 eθx(i) − 1 )β+1 e θx(i)−  λ γ· ( e θx(i) −1 )β · · 1 − 1 − e −  λ γ· ( e θx(i) −1 )β  α − i n + 1  1 − e −  λ γ· ( e θx(i) −1 )β  α−1 (35) ∂ℓ ∂β = −2α · n∑ i=1  λ( eθx(i) − 1 ) γ β ln  λ( eθx(i) − 1 ) γ  · e −  λ( e θx(i) −1 ) γ β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  (36) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3869 ∂ℓ ∂α = −2 n∑ i=1 1 − e − ( λ (eθx−1)γ )β α ln 1 − e − ( λ (eθx−1)γ )β · 1 − 1 − e − ( λ (eθx−1)γ )β α − i n + 1  (37) The O.LS estimates for the parameters Θ can be obtained by minimizing the equation (32) concerning the Θ = (λ, γ, θ, β, α) or by solving the equations from (33) to (37) using numerical techniques available in statistical software. 5.3. Weighted Least Square Method (W.LS) The W.LS method is similar to the O.LS method, which depends on the differences between the empirical and theoretical of the cdf, in addition to the variance of the order statistic as a wight [33]. Therefore, the W.LS function for EFG is written as U2(Θ) = n∑ i=1 ωi [ 1 − [ 1 − exp { − ( λ γ(exθ − 1) )β }]α − ( i n + 1 )]2 (38) where ωi = (n+1)2(n+2) i(n+1−i) The equation (38) is derived for each parameter in the EFG distribution and the derivation equations are given as follows. The W.LS estimation is obtained by minimizing the equation (38) or by solving the nonlinear equations from (39) to (43) using numerical iterative technique available in any statistical software. ∂ℓ ∂λ = −2αβλ(β−1) γβ · n∑ i=1 ωi ·  1( eθx(i) − 1 ) β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  · e −  λ( e θx(i) −1 ) γ β (39) ∂ℓ ∂γ = 2αβλβ γ(β+1) · n∑ i=1 ωi ·  1( eθx(i) − 1 ) β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3870 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  · e −  λ( e θx(i) −1 ) γ β (40) ∂ℓ ∂θ = 2αβ ( λ γ )β · n∑ i=1 ωi · x(i) · ( 1 eθx(i) − 1 )β+1 e θx(i)−  λ γ· ( e θx(i) −1 )β · · 1 − 1 − e −  λ γ· ( e θx(i) −1 )β  α − i n + 1  1 − e −  λ γ· ( e θx(i) −1 )β  α−1 (41) ∂ℓ ∂β = −2α · n∑ i=1 ωi ·  λ( eθx(i) − 1 ) γ β ln  λ( eθx(i) − 1 ) γ  · e −  λ( e θx(i) −1 ) γ β 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 (42) ∂ℓ ∂α = −2 n∑ i=1 ωi · 1 − e −  λ( e θx(i) −1 ) γ β  α ln 1 − e −  λ( e θx(i) −1 ) γ β  · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − i n + 1  (43) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3871 5.4. Cramér-von Mises Method (CRM) [27] introduced the CRM method for estimation parameters. The function of the CRM method for EFG distribution is presented in equation (44). CRM(Θ) = 1 12n + n∑ i=1 [ 1 − [ 1 − exp { − ( λ γ(exθ − 1) )β }]α − (2i − 1 2n )]2 (44) The equations below show the partial derivatives of the parameters Θ = (λ, γ, θ, β, α) from (44). The CRM estimates for EFG parameters are derived by solving the equations from (45) to (49) using numerical technique or by minimizing (44) using optimization technique available in R Package. ∂ℓ ∂λ = −2αβλ(β−1) γβ · n∑ i=1  1( eθx(i) − 1 ) β 1 − e −  λ( e θx(i) −1 ) γ β  α−1 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − 2i − 1 2n  e −  λ( e θx(i) −1 ) γ β (45) ∂ℓ ∂γ = 2αβλβ γ(β+1) · n∑ i=1  1( eθx(i) − 1 ) β · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − 2i − 1 2n  e −  λ( e θx(i) −1 ) γ β (46) ∂ℓ ∂θ = 2αβ ( λ γ )β · n∑ i=1 x(i) · ( 1 eθx(i) − 1 )β+1 e θx(i)−  λ γ· ( e θx(i) −1 )β I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3872 · 1 − 1 − e −  λ γ· ( e θx(i) −1 )β  α − 2i − 1 2n  · 1 − e −  λ γ· ( e θx(i) −1 )β  α−1 (47) ∂ℓ ∂β = −2α · n∑ i=1  λ( eθx(i) − 1 ) γ β ln  λ( eθx(i) − 1 ) γ  e −  λ( e θx(i) −1 ) γ β · 1 − 1 − e −  λ( e θx(i) −1 ) γ β  α − 2i − 1 2n  · 1 − e −  λ( e θx(i) −1 ) γ β  α−1 (48) ∂ℓ ∂α = −2 n∑ i=1 1 − e − ( λ (eθx−1)γ )β α ln 1 − e − ( λ (eθx−1)γ )β 1 − 1 − e − ( λ (eθx−1)γ )β α − 2i − 1 2n  (49) 5.5. Maximum product of spacing Method (MPS) [12] proposed the MPS method in order to improve the performance of the MLH estimator. Let a random sample from EFG distribution with size n and x(1), x(2), ..., x(n) is the corresponding ordered sample. The idea of the MPS method is to optimize the geometric mean of spacings, which refers to the variations between the CDF values of adjacent data points. The spacings between neighboring ordered values can be defined as Di(Θ) = F (x(i); Θ) − F (x(i−1); Θ), Θ = (λ, γ, θ, β, α), i = 1, . . . , n + 1. Therefore, the MPS function for EFG is given as M(Θ) = 1 n + 1 n+1∑ i=1 logDi(Θ) M(Θ) = 1 n + 1 n+1∑ i=1 log [1 − exp − ( λ γ(eθx(i−1) − 1) )β  α − 1 − exp − ( λ γ(eθx(i) − 1) )β  α ] (50) The first partial derivative of (50) with respect to Θ = (λ, γ, θ, β, α), are given as follows. The MPS estimates for EFG parameters are derived by solving the equations I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3873 from (51) to (55) using numerical technique or via maximizing (50) using optimization technique available in R Package. ∂ℓ ∂λ = αβλ(β−1) γβ · (n + 1) · n+1∑ i=1  1 − e −  λ( e θx(i−1) −1 ) γ β  α − 1 − e −  λ( e θx(i) −1 ) γ β  α −1 · ·  ( 1 eθx(i)−1 − 1 )β e −  λ( e θx(i)−1 −1 ) γ β 1 − e −  λ( e θx(i)−1 −1 ) γ β  α−1 −  ( 1 eθx(i−1) − 1 )β e −  λ( e θx(i−1) −1 ) γ β 1 − e −  λ( e θx(i−1) −1 ) γ β  α−1 (51) ∂ℓ ∂γ = αβλβ γ(β+1) · (n + 1) · n+1∑ i=1  1 − e −  λ( e θx(i−1) −1 ) γ β  α − 1 − e −  λ( e θx(i) −1 ) γ β  α −1 ·  ( 1 eθx(i) − 1 )β e −  λ( e θx(i) −1 ) γ β 1 − e −  λ( e θx(i) −1 ) γ β  α−1 −  ( 1 eθx(i−1) − 1 )β e −  λ( e θx(i−1) −1 ) γ β 1 − e −  λ( e θx(i−1) −1 ) γ β  α−1 (52) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3874 ∂ℓ ∂θ = αβλβ (n + 1) · γβ · n+1∑ i=1  1 − e −  λ( e θx(i)−1 −1 ) γ β  α − 1 − e −  λ( e θx(i−1)−1 −1 ) γ β  α −1 ·  x(i−1) · ( 1 eθx(i−1) − 1 )β+1 · e θx(i−1)−  λ γ· ( e θx(i−1) −1 )β · 1 − e −  λ γ· ( e θx(i−1) −1 )β  α−1 −  x(i) · ( 1 eθx(i) − 1 )β+1 · e θx(i)−  λ γ· ( e θx(i) −1 )β · 1 − e −  λ γ· ( e θx(i) −1 )β  α−1 (53) ∂ℓ ∂β = α (n + 1) · n+1∑ i=1  1 − e −  λ( e θx(i−1)−1 −1 ) γ β  α − 1 − e −  λ( e θx(i)−1 −1 ) γ β  α −1 · 1 − e −  λ( e θx(i)−1 −1 ) γ β  α−1 ·  λ( eθx(i)−1 − 1 ) γ β ·ln  λ( eθx(i)−1 − 1 ) γ ·e −  λ( e θx(i)−1 −1 ) γ β − 1 − e −  λ( e θx(i−1)−1 −1 ) γ β  α−1 ·  λ( eθx(i−1)−1 − 1 ) γ β ·ln  λ( eθx(i−1)−1 − 1 ) γ ·e −  λ( e θx(i−1)−1 −1 ) γ β (54) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3875 ∂ℓ ∂α = 1 (n + 1) · n+1∑ i=1  1 − e −  λ( e θx(i−1)−1 −1 ) γ β  α − 1 − e −  λ( e θx(i)−1 −1 ) γ β  α −1 · ln 1 − e −  λ( e θx(i)−1 −1 ) γ β  · 1 − e −  λ( e θx(i)−1 −1 ) γ β   − ln 1 − e −  λ( e θx(i−1)−1 −1 ) γ β  · 1 − e −  λ( e θx(i−1)−1 −1 ) γ β   (55) 6. Numerical Study The Monte Carlo simulation technique is carried out to assess and compare the effec- tiveness of the various estimation methods that are used for estimating the parameters of EFG. The simulation involved using gradient samples of varying sizes, beginning with a small size of 15 and gradually increasing to larger sizes of 30, 50, 100, and 200. Further- more, two sets of parameters have been implemented: Set I: (λ=2.4, γ=0.5, θ=0.3, β=0.7, α=1.2). Set II: (λ=0.5, γ=3, θ=0.7, β=1.5, α=2.4). The bias and mean square error (MSE) of the parameter estimation are calculated for each simulation scenario. Tables 1 and 2 show the simulation results. The table displays the parameter estimation values and corresponding bias and MSE for each method of estimation. It appears that with an increase in the sample size, the MSE values for all the estimators decrease. Out of all the estimators, the MLH estimator has the lowest MSE value. Based on the Figures 2 and 5, it seems that MLH estimator is the closest estimate of the parameter at different sample sizes, while the other estimators get closer to the correct pa- rameter values as the sample size increases. Additionally, the figures show that the MPS estimate is the farthest from the actual value for certain parameters at smaller sample sizes. However, all estimators are equally accurate at a large sample size of 200. Based on Figures 3, 4, 6 and 7. It seems that as the sample size increases, the MSE values for all estimators decrease. The MLH estimator appears to have the least MSE compared to the other estimators. For smaller sample sizes, the MSE values for the es- timators O.LS, W.LS and CRM are quite similar. However, the MPS estimator has the highest MSE values compared to the other estimators, particularly for smaller sample sizes. I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3876 Table 1: Parameter estimation from five different methods (Set I) Set I (λ=2.4, γ=0.5, θ=0.3, β=0.7, α=1.2) n Par. MLH O.LS W.LS CRM PMS 15 λ̂ 2.3512 2.3514 2.2721 2.2945 2.7462 Bias 0.0880 1.3073 1.3129 1.4186 2.2728 MSE 0.1449 2.4218 2.4173 6.1409 5.2112 γ̂ 0.5853 0.7343 0.7397 0.5872 0.8009 Bias 0.3438 0.6234 0.6168 0.5049 0.8180 MSE 0.4272 0.9822 1.0735 0.7979 1.9747 θ̂ 0.4146 0.4041 0.4078 0.4029 0.5314 Bias 0.1279 0.2022 0.2041 0.1879 0.3605 MSE 0.1874 0.2770 0.2781 0.2570 0.4948 β̂ 1.2062 1.3493 1.3998 1.4720 1.4974 Bias 0.5773 0.8197 0.8667 0.9135 1.0597 MSE 0.7961 1.3511 1.4236 1.4539 1.9745 α̂ 1.0688 1.5481 1.5404 1.8106 2.2968 Bias 0.4159 0.9719 1.0091 1.2483 1.8040 MSE 0.4933 1.6519 1.6428 5.8150 6.9145 30 λ̂ 2.3543 2.1697 2.2125 2.1767 2.2907 Bias 0.0796 0.8750 1.0115 0.9276 1.4084 MSE 0.1150 1.4818 2.8512 1.5135 2.2168 γ̂ 0.6052 0.6801 0.6010 0.5952 0.6313 Bias 0.3305 0.5076 0.4423 0.4615 0.5785 MSE 0.3994 0.8231 0.6472 0.7740 1.1232 θ̂ 0.3975 0.3724 0.3965 0.3889 0.5049 Bias 0.1080 0.1486 0.1725 0.1577 0.3068 MSE 0.1655 0.1973 0.2327 0.2131 0.4302 β̂ 1.1226 1.2623 1.2033 1.2796 1.1178 Bias 0.4793 0.6863 0.6331 0.7080 0.6009 MSE 0.6604 1.1075 0.9828 1.1464 1.0554 α̂ 1.0755 1.4750 1.4975 1.6354 1.5666 Bias 0.3603 0.8252 0.8498 0.9924 0.9858 MSE 0.4352 1.3811 1.6379 1.7390 1.6794 50 λ̂ 2.3654 2.1632 2.1517 2.1655 2.3983 Bias 0.0629 0.6827 0.6992 0.6994 1.2418 continued on next page I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3877 Table1 :Parameter estimation...(Set I) – (continue) Set I (λ=0.5, γ=3, θ=0.7, β=1.5, α=2.4) n Par. MLH O.LS W.LS CRM PMS MSE 0.0858 1.2851 1.1158 1.4107 2.1186 γ̂ 0.5890 0.6261 0.5891 0.6034 0.6500 Bias 0.2698 0.4180 0.3784 0.3881 0.5074 MSE 0.3298 0.5994 0.5783 0.6171 0.9859 θ̂ 0.3767 0.3706 0.3849 0.3767 0.4654 Bias 0.0824 0.1319 0.1398 0.1322 0.2635 MSE 0.1319 0.1778 0.1966 0.1807 0.3756 β̂ 1.0309 1.1495 1.1011 1.1891 1.0577 Bias 0.3635 0.5474 0.4859 0.5680 0.4699 MSE 0.4649 0.8643 0.7305 0.8702 0.7161 α̂ 1.1120 1.3688 1.3172 1.3952 1.4319 Bias 0.2647 0.6773 0.6025 0.7374 0.8438 MSE 0.3332 1.0368 0.8795 1.1953 1.4303 100 λ̂ 2.3723 2.1838 2.2314 2.1722 2.3261 Bias 0.0497 0.4811 0.4522 0.4706 0.8865 MSE 0.0661 0.7516 0.7180 0.7981 1.5979 γ̂ 0.5763 0.6084 0.5951 0.5827 0.5667 Bias 0.2128 0.3248 0.2888 0.3086 0.3828 MSE 0.2574 0.5113 0.4116 0.4396 0.8550 θ̂ 0.3623 0.3611 0.3640 0.3662 0.4438 Bias 0.0653 0.1030 0.0999 0.1063 0.2198 MSE 0.0985 0.1401 0.1406 0.1460 0.3195 β̂ 0.9861 1.0927 1.0592 1.1041 0.9392 Bias 0.2997 0.4475 0.3985 0.4560 0.3131 MSE 0.3687 0.6403 0.5507 0.6640 0.4185 α̂ 1.1205 1.2691 1.2199 1.2531 1.3295 Bias 0.1763 0.5499 0.4654 0.5245 0.6425 MSE 0.2299 0.8246 0.6782 0.7788 0.9742 200 λ̂ 2.3693 2.2044 2.2811 2.1933 2.399 Bias 0.0470 0.3583 0.3040 0.3442 0.7237 MSE 0.0602 0.5660 0.4806 0.5366 1.4105 γ̂ 0.6100 0.6133 0.6261 0.6138 0.5544 Bias 0.2043 0.2594 0.2363 0.2608 0.2879 MSE 0.2459 0.3524 0.3196 0.3639 0.4193 θ̂ 0.3482 0.3466 0.3416 0.3446 0.4114 Bias 0.0508 0.0814 0.0686 0.0795 0.1728 MSE 0.0771 0.1127 0.0979 0.1092 0.2581 continued on next page I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3878 Table1 :Parameter estimation...(Set I) – (continue) Set I (λ=0.5, γ=3, θ=0.7, β=1.5, α=2.4) n Par. MLH O.LS W.LS CRM PMS β̂ 0.9965 1.0396 1.0239 1.0433 0.9337 Bias 0.3015 0.3651 0.3361 0.3671 0.2667 MSE 0.3472 0.4695 0.4147 0.4762 0.3266 α̂ 1.1079 1.1799 1.1661 1.2015 1.2353 Bias 0.1660 0.3560 0.3165 0.3699 0.4783 MSE 0.2057 0.4830 0.4182 0.5231 0.6832 Table 2: Parameter estimation from five different methods (Set II) Set II (λ=2.4, γ=0.5, θ=0.3, β=0.7, α=1.2) n Par. MLH O.LS W.LS CRM PMS 15 λ̂ 1.3173 1.1633 1.2466 1.3151 1.3479 Bias 0.9575 1.1856 1.2501 1.2147 1.6603 MSE 1.3253 2.6838 2.6338 2.4629 4.3668 γ̂ 2.5823 3.3157 3.1059 2.9964 3.1173 Bias 0.4383 2.0975 1.9570 2.0707 2.5715 MSE 0.7287 3.6010 3.1051 3.8074 5.3895 θ̂ 1.4061 0.9920 1.0960 1.1645 1.2406 Bias 0.8966 0.9272 0.9912 0.9403 1.2764 MSE 1.2245 1.7337 1.7678 1.5917 2.0885 β̂ 1.3022 1.4707 1.4469 1.5112 1.6331 Bias 0.5074 0.9303 0.9227 0.9219 1.1114 MSE 0.6403 1.3643 1.3735 1.3535 1.9297 α̂ 2.2546 3.7934 3.9230 4.2419 5.9237 Bias 1.0808 2.7599 2.8551 3.1269 5.0923 MSE 1.2991 6.1538 6.5440 6.8010 18.1692 30 λ̂ 1.1721 1.1353 1.1595 1.2092 0.9323 Bias 0.7999 0.9932 1.0928 1.0549 1.0962 MSE 1.0942 2.1260 2.2174 2.1770 2.4700 γ̂ 2.6875 3.0399 3.1581 2.9533 3.2234 Bias 0.3306 1.5615 1.6747 1.6252 2.1068 MSE 0.5654 2.2631 2.6847 3.7378 4.2600 θ̂ 1.1944 0.9915 0.9529 1.0952 0.9429 continued on next page I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3879 Table2 :Parameter estimation...(Set II) – (continue) Set II (λ=2.4, γ=0.5, θ=0.3, β=0.7, α=1.2) n Par. MLH O.LS W.LS CRM PMS Bias 0.7035 0.7708 0.8299 0.8488 0.9901 MSE 0.9602 1.4288 1.3899 1.4607 1.6179 β̂ 1.1308 1.2555 1.2452 1.2808 1.3091 Bias 0.4412 0.7083 0.7196 0.7061 0.7921 MSE 0.5113 0.9514 0.9675 0.8969 1.2750 α̂ 2.2898 3.5279 3.8083 3.5672 4.9085 Bias 0.8181 2.3179 2.5358 2.3424 3.7576 MSE 1.0171 5.2966 6.4424 4.4101 13.8399 50 λ̂ 1.1081 1.0230 0.9116 1.0299 0.7250 Bias 0.7285 0.8321 0.8173 0.8375 0.7688 MSE 1.0041 1.6203 1.6048 1.6681 1.8030 γ̂ 2.7400 3.1485 3.0437 2.9673 3.1842 Bias 0.2777 1.3605 1.3479 1.3011 1.8411 MSE 0.4723 1.9355 2.0288 1.7906 3.1472 θ̂ 1.0677 0.8891 0.8341 0.9384 0.7971 Bias 0.5804 0.6812 0.6886 0.6898 0.8131 MSE 0.7856 1.1375 1.0790 1.1663 1.6433 β̂ 1.0720 1.1172 1.0975 1.1837 1.1255 Bias 0.4519 0.5965 0.5688 0.5880 0.6143 MSE 0.5018 0.6921 0.6572 0.7098 0.7777 α̂ 2.3206 3.2388 3.2671 3.2239 4.0809 Bias 0.6817 1.8042 1.7741 1.8734 2.7098 MSE 0.8571 3.0798 3.1481 3.7484 7.1000 100 λ̂ 0.9495 0.8179 0.7823 0.8777 0.6716 Bias 0.5370 0.5686 0.5446 0.6090 0.6284 MSE 0.7206 1.0441 0.9622 1.1496 1.1765 γ̂ 2.8457 3.0526 3.0523 3.0116 3.3954 Bias 0.1683 1.0339 0.9871 1.0464 1.6882 MSE 0.2711 1.3585 1.3000 1.3737 2.8454 θ̂ 0.9178 0.7481 0.7597 0.8205 0.6769 Bias 0.4089 0.4939 0.5212 0.5410 0.6537 MSE 0.5590 0.7567 0.7646 0.8684 1.0054 β̂ 1.0485 1.0705 1.0508 1.0858 1.0268 Bias 0.4562 0.5283 0.05091 0.5250 0.5384 MSE 0.4870 0.5948 0.5585 0.5901 0.6105 α̂ 2.3061 3.0350 2.8741 2.9541 3.4503 Bias 0.5229 1.4461 1.2220 1.3575 1.8577 continued on next page I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3880 Table2 :Parameter estimation...(Set II) – (continue) Set II (λ=2.4, γ=0.5, θ=0.3, β=0.7, α=1.2) n Par. MLH O.LS W.LS CRM PMS MSE 0.6534 2.5415 1.9669 2.2529 2.8001 200 λ̂ 0.9068 0.7664 0.7823 0.7629 0.6124 Bias 0.4635 0.4518 0.5446 0.4391 0.4917 MSE 0.6317 0.6975 0.9622 0.6648 0.8302 γ̂ 2.8778 3.0805 3.0523 3.0105 3.4361 Bias 0.1324 0.8169 0.9871 0.7978 1.3674 MSE 0.2184 1.0906 1.3000 1.0826 2.0207 θ̂ 0.8507 0.7006 0.7597 0.7193 0.5915 Bias 0.3253 0.4022 0.5212 0.3869 0.5609 MSE 0.4267 0.5675 0.7646 0.5341 0.8099 β̂ 1.0180 1.0246 1.0508 1.0351 0.9873 Bias 0.4821 0.5069 0.5091 0.4959 0.5298 MSE 0.4991 0.5511 0.5585 0.5407 0.5694 α̂ 2.3671 2.8549 2.8741 2.8335 3.3185 Bias 0.4047 1.0719 1.2220 1.0796 1.5162 MSE 0.5222 1.5927 1.9669 1.6631 2.1942 I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3881 Figure 2: Compare parameter estimation from different methods at each sample size set I I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3882 Figure 3: Compare MSE for different estimation methods at each sample size set I I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3883 Figure 4: MSE at different n set I I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3884 Figure 5: Compare parameter estimation from different methods at each sample size set II I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3885 Figure 6: Compare MSE for different estimation methods at each sample size set II I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3886 Figure 7: MSE at different n set II I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3887 7. Application In this section, the newly developed distribution is applied to real-world data from patients with various types of cancer, including brain, bladder, Bone and blood cancers. The analysis involves comparing different estimation methods, as well as evaluating the performance of the new distribution against other competing distributions. 7.1. Compare between estimation methods In this section, different datasets of cancer patient data are analyzed, each group suffering from a different type of cancer such as bladder cancer, brain cancer, bone cancer and blood cancer . The data are modeled using the EFG model and the model parameters are estimated using the five different estimation methods. The datasets are provided below. Dataset I: Bladder Cancer The remission times by months of bladder cancer patients were recorded in a dataset that included 36 individuals [20]. Dataset II: Brain Cancer This data represents the lifetime by day for 87 patients with brain cancer diseases collected from the Atomic Medicine and Radiance Hospital in Baghdad [22]. Dataset III: Bone Cancer Simulated data has been considered by [29] to represent the survival times (in days) of 73 patients who have been diagnosed with acute bone cancer. Dataset IV: Blood Cancer This data consists of the lifetime (in years) of 40 blood cancer (leukemia) patients from one of the Ministry of Health hospitals in Saudi Arabia reported in [6] Dataset V: Bladder Cancer This data represents the duration of remission (in months) for a group of 128 patients who were diagnosed with bladder cancer [26]. Table 3: Descriptive statistics for the datasets Data Min Q1 Median Mean Q3 Max Skw Kur Dataset I 0.08 1.16 2.08 1.94 2.71 3.36 -0.29 1.86 Dataset II 11 37.5 67.5 78.26 92 274 1.47 5.37 Dataset III 0.09 0.92 1.57 3.76 2.75 86.01 6.79 51.78 Dataset IV 0.32 2.19 3.35 3.14 4.26 5.38 -0.42 2.27 Dataset V 0.08 3.35 6.39 9.37 11.84 79.05 3.29 18.48 Table 3 displays a summary for the datasets. The table presents a range of skewness and kurtosis values that reflect different shapes of the data distribution. The skewness values vary in terms of direction and severity, with some being positive and others negative, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3888 and some being large while others are small. Similarly, the kurtosis values also differ, with some being small and others large, indicating variations in the width of the distribution’s tail. Table 4: Estimation Survival time for bladder cancer patients (Dataset I) Est. α̂ θ̂ γ̂ λ̂ β̂ KS P.value MLH 7.2725 0.2072 9.8959 0.0040 1.9577 0.1467 0.4208 (5.7753) (0.0587) (26.4236) (0.0031) (0.7532) O.LS 6.3994 0.2570 2.1266 0.0068 1.3603 0.1487 0.4036 (29.7235) (0.3706) (20.2968) (0.0188) (3.6681) W.LS 3.0007 0.2090 5.7650 0.0048 2.6659 0.1442 0.4422 (0.6459) (0.0142) (2.4854) (0.0012) (0.3396) MPS 1.7373 0.2571 2.1588 0.0152 2.5544 0.1554 0.3493 (0.9347) (0.0862) (9.3294) (0.0699) (0.9404) CVM 1.0113 0.5146 0.0609 0.0050 1.7158 0.1670 0.2681 (3.2626) (1.9698) (0.7743) (0.0114) (6.1166) Figure 8: Estimation Survival time for Bladder cancer patients (Dataset I) I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3889 Table 5: Estimation Survival time for Brain cancer patients Est. α̂ θ̂ γ̂ λ̂ β̂ KS P.value MLH 0.6469 0.5550 0.3064 0.0390 0.0560 0.1244 0.4211 (0.6738) (0.2586) (2.0015) (0.2472) (0.0528) O.LS 0.0685 0.1171 1.3718 2.4640 1.3474 0.1454 0.2410 (0.2126) (0.3623) (0.5018) (0.5015) (0.1552) W.LS 0.0832 0.0502 2.3820 1.1014 2.5989 0.1459 0.2373 (0.0081) (0.0050) (0.1833) (0.3195) (0.0910) MPS 0.4982 0.4982 0.3169 0.0325 0.0729 0.1213 0.4542 (0.4306) (0.2841) (2.1811) (0.2197) (0.0659) CVM 0.0460 0.0776 1.5981 0.6614 3.0128 0.1497 0.2122 (0.1406) (0.2408) (0.0884) (0.0885) (0.1441) Figure 9: Estimation survival time for brain cancer patients . I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3890 Table 6: Estimation survival time for acute bone cancer patients Est. α̂ θ̂ γ̂ λ̂ β̂ KS P.value MLH 3.7435 0.5731 0.0898 2.5659 0.0080 0.0908 0.5843 (2.0732) (0.1339) (0.7592) (20.2149) (0.0042) O.LS 1.2000 0.7829 0.2822 0.2372 0.5734 0.0683 0.8852 (28.0826) (5.7028) (14.3646) (10.5554) (10.3191) W.LS 1.8908 0.7619 0.0752 0.1117 0.2702 0.0714 0.8503 ( 0.4005) (0.0941) (0.0767) ( 0.0640) (0.1330) MPS 1.1891 0.8510 0.0123 1.1132 0.0092 0.1330 0.1510 (0.3298) (0.1210) ( 0.0131) ( 1.3190) (0.0051) CVM 1.2552 0.8531 0.0054 0.0065 0.4297 0.0702 0.8649 (7.5602) (1.7563) (0.0052) (0.0087) (1.9309) Figure 10: Estimation Survival time for acute bone cancer patients I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3891 Table 7: Estimation survival time for blood cancer patients Est. α̂ θ̂ γ̂ λ̂ β̂ KS P.value MLH 12.5138 0.2339 11.5206 0.0044 1.0836 0.1104 0.7143 (10.2835) (0.0876) (39.7697) (0.0026) (0.4653) O.LS 0.3832 0.7476 0.4354 0.0084 2.01428 0.1403 0.4103 (1.7182) (3.4603) (4.1605) (0.0570) (6.5835) W.LS 1.6424 0.2544 1.1654 0.0005 2.4439 0.1012 0.8070 (0.3505) (0.0263) (0.4141) (0.0003) (0.0797) MPS 1.5480 0.3466 1.0598 0.0139 1.4797 0.1399 0.4139 (0.8931) (0.1544) (3.1758) (0.0613) (0.6700) CVM 2.0440 0.2554 1.8712 0.0009 2.1143 0.1010 0.8092 (10.1339) (0.4582) (16.1734) (0.0042) (0.9868) Figure 11: Estimation Survival time for blood cancer patients I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3892 Table 8: Estimation survival time for bladder cancer patients (Dataset V) Est. α̂ θ̂ γ̂ λ̂ β̂ KS P.value MLH 0.2559 0.4308 0.1958 0.0647 1.0659 0.0708 0.5419 (0.0229) (0.0036) (0.0036) (0.0036) (0.0036) O.LS 0.1478 0.6019 1.5195 0.3035 1.4830 0.0321 0.9994 (0.0747) (0.1446) (4.3697) (0.1478) (0.1464) W.LS 2.2327 0.5686 0.0349 0.0313 0.0842 0.0349 0.9977 (0.2789) (0.0255) (0.0055) (0.0050) (0.0120) MPS 0.2632 0.4361 0.6715 0.2136 1.0025 0.0831 0.3400 (0.0287) (0.0136) (0.0150) (0.0148) (0.0646) CVM 0.9312 0.7291 0.1447 0.1240 0.1832 0.0370 0.9948 (4.8391) (1.7438) (3.2033) (2.4158) (0.8814) Figure 12: Estimation Survival time for bladder cancer patients (Dataset V) The Tables from 4 to 8 display the results of parameter estimation and its standard error (SE) (in parentheses). The tables also include the test statistic of the Kolmogorov Smirnov (KS) goodness of fit test along with its P.value which is used to check the fitting of data with the EFG distribution. The estimated distribution with the lowest KS and highest p.value is considered as the best fit for data. Additionally, to compare the fitting of the data with the estimated EFG distribution by the estimation methods, the figures from 8 to 12 display the empirical distribution of each dataset concurrently with the estimated pdf and cdf of EFG distribution by the five estimation methods. Based on the results presented in the tables from 4 to 8, it seems that the EFG distribution could effectively represent the survival time data for cancer patients since the I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3893 p.values are greater than 0.05 in all of the six datasets. Furthermore, the W.LS method provides the best estimation for EFG parameters fitting with the bladder cancer datasets (Table 4 and Table 8). Even though, the CVM method produces the highest p.value for representing the bone cancer and blood cancer datasets as seen in Tables 6 and 7, the W.LS still provides a close fit to the data with a smaller SE comparing to CVM. According to brain cancer and breast cancer datasets, the MPS method gives the best fitting to these datasets. 7.2. Compare EFG against other distributions This section provides a comparison of the performance of the EFG model against four competing distributions, each with a different number of parameters. The CDFs of these distributions are outlined below. • Exponentiated Gompertz Exponential (EGoG) [1] F (x) = { 1 − e γ θ [1−eλxθ] }α , x > 0; θ, γ, λ, α > 0. • The Fréchet F (x) = exp { − ( x−λ θ )−α } , x > 0; α, θ > 0, −∞ < λ < ∞. • Gompertz given in (3). • The exponential(E) F (x) = θ e−θx,. The EFG model is evaluated using several goodness-of-fit indices (GoF). These include the negative log-likelihood value (−ℓ), Akaike Information Criterion (AIC), corrected AIC (CAIC) and Bayesian Information Criterion (BIC). Kramér-von Mises (KVM) test statis- tic, Kolmogorov-Smirnov (KS) test statistic, and their corresponding p-values are calcu- lated as well. A model is deemed the best fit for the data when its values for these statistics are lower than those of the competing models. The tables from (9) to (12) present the GoF measurements for the EFG model and the competing distributions. The results indicate that the exponential and Gompertz distributions did not adequately fit all the data, as they yielded significant p-values less than 0.05. In contrast, other distributions, such as EGoG and Fréchet, demonstrated p-values greater than 0.05 for most datasets, suggesting they can effectively represent the data. While these models fit the majority of datasets (p-value> 0.05), the EFG model exhibited the highest p-value among all fitted models. Additionally, the tables show that the NEG distribution achieved the lowest values across all GoF indices, indicating that it fits all five datasets better than the other competing distributions. I. A. Alsaggaf / Eur. J. Pure Appl. Math, 17 (4) (2024), 3856-3898 3894 Table 9: The Performance of EFG for Bladder cancer data. Distributions EFG EGoE Fréchet Gompertz E −ℓ 49.481 72.603 53.814 66.089 59.857 AIC 108.962 153.206 113.628 136.179 121.714 CAIC 113.529 156.374 116.004 137.762 122.505 BIC 116.879 159.541 118.379 139.346 123.297 KVM 0.126 0.437 0.159 1.089 0.656 p-value 0.473 0.057 0.137 0.001 0.016 KS 0.146 0.190 0.175 0.333 0.230 p-value 0.428 0.147 0.222 0.001 0.044 Table 10: The Performance of EFG for Brain cancer data. Distributions EFG EGoE Fréchet gompertz E −ℓ 260.789 302.709 270.203 266.799 268.003 AIC 531.578 613.417 546.405 537.599 538.006 CAIC 536.358 617.241 549.273 539.511 549.273 BIC 541.138 621.065 552.141 541.423 539.918 KVM 0.089 6.290 0.858 0.556 0.522 p-value 0.646 2.2x10−16 0.005 0.028 0.035 KS 0.124 0.558 0.284 0.226 0.186 p-value 0.421 5.9x10−14 0.001 0.012 0.062 Table 11: The Performance of EFG for Bone cancer data. Distributions EFG EGoE Fréchet gompertz E −ℓ 142.975 146.544 155.419 203.825 169.589 AIC 295.951 301.088 316.839 411.651 341.179 CAIC 301.677 305.669 320.275 413.941 320.275 BIC 307.403 310.249 323.710 416.232 343.4693 KVM 0.139 0.172 0.222 0.423 1.546 p-value 0.426 0.329 0.229 0.062 1.2x10−4 KS 0.103 0.091 0.1219 0.182 0.251 p-value 0.584 0.379 0.2281 0.016 1.9 x10−4 REFERENCES 3895 Table 12: The Performance of EFG for Blood cancer data. Distributions EFG EGoE Fréchet gompertz E −ℓ 68.592 75.771 74.386 83.953 85.778 AIC 147.185 159.542 154.772 171.9054 173.556 CAIC 151.407 162.920 157.306 173.594 174,401 BIC 155.629 166.298 159.839 175.283 175.245 KVM 0.094 0.221 0.201 1.077 1.084 p-value 0.616 0.230 0.267 0.001 0.001 KS 0.110 0.139 0.136 0.299 0.300 p-value 0.714 0.421 0.451 0.002 0.001 8. Conclusion Obtaining accurate estimates regarding patients’ condition is extremely important as this helps the doctor determine the appropriate treatment. This study focused on inves- tigating the EFG distribution and its effectiveness in representing the survival time of cancer patients. It also provides a comprehensive overview of the statistical properties of the distribution such as the quantile function,moments, moment-generating function, characteristic function, R´enyi entropy, and order statistics More addition, the study also examined the performance of five different types of estimates, MLH, O.LS, W.LS, CRM and MPS including simulation and application to real data for cancer patients. The re- sults revealed that the EFG distribution is an ideal representation of the survival time for patients with various types of cancer. Additionally, the W.LS method provided the best estimate for survival time with minimal SEs. Moreover, the performance of the EFG model is assessed in comparison to various competing distributions. Notably, EFG proves to be a more suitable choice than several of these models, as it consistently achieves the lowest values across multiple goodness-of-fit criteria. 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