EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6121 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Novel Generalization of the Inverted Nadarajah–Haghighi Distribution: Estimation Methods and Medical Applications Sara F. Aloufi1, Ibtesam A. Alsaggaf1,∗ 1 Department of Statistics, Faculty of Science, King Abdul-Aziz University, Jeddah, 21589, Saudi Arabia Abstract. Statisticians and data analysts often use statistical probability distributions to de- scribe and analyze their data. However, traditional distributions may not always accommodate certain data sets, making it necessary to develop new models to handle complex data struc- tures and improve fit quality. This paper introduces a new extended lifetime model, called the New Exponential Inverted Nadarajah–Haghighi distribution (NEINH), which belongs to the new exponential-X family of distributions. This approach is designed to model complex data in a variety of applications. The article explores some of the statistical properties of this proposed distribution such as quantile function, moment, moment generating function, order statistic and others. The NEINH’s parameters are estimated using Bayesian estimation method and maxi- mum likelihood method and five other methods. The efficacy of this distribution is established by its comparative analysis with alternative distributions, using four real-world medical datasets to highlight its superior performance. 2020 Mathematics Subject Classifications: 60E05, 62E10, 62N05 Key Words and Phrases: T-X family, exponential-X family of distributions, inverted Nadara- jah–Haghighi, maximum likelihood, maximum product of spacing, ordinary and weighted least squares estimators, Bayesian, Anderson–Darling estimators, Monte Carlo simulations 1. Introduction Probability distributions are fundamental for data modeling in numerous fields, such as finance, technology, Biological sciences, industry, and healthcare. Consequently, re- searchers have introduced new extended distributions to enhance the performance of density and hazard rate functions in applications. Methods employed to extend distributions include compounding, parameter addi- tion, composition, and transformation. Examples encompass [1] introduced the beta- generation approach, [2] proposed the Kumaraswamy-generated approach, and [3] de- veloped the transformed-transformer approach, along with several other approaches. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6121 Email address: sfalaufe@kau.edu.sa (S. F. Aloufi), ialsaggaf@kau.edu.sa (I. A. Alsaggaf) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 2 of 29 A novel family of lifetime distributions known as the exponential-X (NLTE-X) in- troduced by [4]. The basis of this distribution family is the T-X generator, where T ∼ Exp(1) and W (G(x)) = − log { 1−G(x) eϑG(x) } . For the NLTE-X family, the CDF and PDF are expressed as follows: F (x; ϑ, ζ) = 1 − {1 − G(x; Ω) eϑG(x;Ω) } ; ϑ > 0, x > 0, (1) f(x; ϑ, ζ) = g(x){1 + ϑG(x; Ω)} eϑG(x;Ω) ; ϑ > 0, x > 0, (2) where ϑ represents the vector of distribution parameters, and the parameter ϑ is unique to the NLTE-X family. Several distributions have been derived from the NLTE-X fam- ily, including the exponential Fréchet distribution [5], exponential inverted Topp–Leone distribution [6], exponentiated Weibull distribution [7], exponential-X power family of distributions [8], the exponential generalized inverse generalized Weibull distribution [9], and the exponential inverted Gompertz distribution [10]. Inverted distributions have attracted significant attention from researchers due to their enhanced flexibility in the structure of both the density and hazard functions compared to their non-inverted counterparts. Furthermore, inverted distributions have proven to be highly applicable in various fields, including biological research, life test- ing problems, chemical data analysis, and technological applications. For instance, the inverted exponential distribution [11] was applied in analyzing failure rates and repair times of computer numerical control machine tools to evaluate their reliability in in- dustrial operations. The inverse Weibull distribution was applied to medical data [12] to model life data with decreasing failure rates and to reliability data [13] for systems with non-monotonic failure rates. The inverse Rayleigh distribution [14] was used to analyze failure times of manufactured components in industrial reliability testing. The inverted Kumaraswamy distribution [15] was applied to reliability data in industrial sys- tems, and the inverted Lindley distribution [16] was used in medical survival data. The inverted Topp-Leone distribution [17] was applied to industrial reliability data, while the inverse Lomax distribution [18] was used in economic data for income and wealth distribution analysis. The inverse power Lomax distribution [19] and the alpha power transformed inverse Lomax distribution [20] were both applied to industrial reliability data in engineering systems. Recently, various generalizations of inverse distributions have been presented in the literature. These include, the inverse Weibull inverse exponential distribution [21], the extended inverse Weibull distribution [22], the the Weibull inverse Rayleigh distribution [23], the Topp–Leone inverted Kumaraswamy distribution [24], the extended inverse Lindley distribution [25], and the odd Weibull inverse Topp–Leone distribution proposed by [26], the Kumaraswamy generalized inverse Lomax distribution [27]. Moreover, a novel inverted model known as the inverted Nadarajah–Haghighi (INH) distribution was introduced by [28], with a decreasing and unimodal density, along with decreasing and upside-down bathtub hazard rate shapes. Several statistical properties of the INH distribution were derived, and various methods were used to estimate the S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 3 of 29 model’s parameters. The suitability of the INH distribution has been demonstrated by testing it on real-life datasets. The CDF and PDF of INH are represented by GINH(x) = e { 1− ( 1+ κ x )τ} ; x > 0, τ, κ > 0, (3) gINH(x) = τκx−2 ( 1 + κ x )τ−1 e { 1− ( 1+ κ x )τ} ; x > 0, (4) where τ is the shape parameter and κ is the scale parameter.In addition, estimators for the parameters of the INH distribution were derived by [29] using several methods, including maximum likelihood estimation, Bayesian estimation, and the Maximum Prod- uct of Spacing approach. Moreover, the generalization of the INH distribution has been extensively investigated by various researchers, these include, the Marshall-Olkin INH distribution [30], extended odd Weibull INH distribution [31], power INH distribution [32], odd Lomax INH distribution [33], and Half Logistic INH distribution [34]. The primary aim of this article is to introduce a novel generalization of INH distribu- tion which is derived from the NLTE-X family of distributions called the new exponential inverted Nadarajah-Haghighi (NEINH) distribution. The proposed distribution is ex- pected to enhance the characteristics and flexibility of the density and hazard rate func- tions. The adaptability of the NEINH distribution is investigated to describe survival time by analysing several medical datasets. Moreover, the primary reason for employing NEINH in practice is to increase INH’s flexibility by introducing new generalizations and providing a superior fit compared to other competing models. This article is organized as follows: Section 2 introduces the NEINH model along with graphical representations. Section 3 discusses the derived properties of the NEINH. Section 4, seven estimation methods are employed to estimate the NEINH parameter, including maximum likelihood (ML), maximum product of spacing (MPS), Bayesian, ordinary least squares (OLS), weighted least squares (WLS), Cramér–von Mises (CM), and Anderson–Darling (AD). Section 5 presents extensive simulation studies to evaluate the performance of these estimators. Section 6 covers four applications in medicine. Finally, Section 7 provides concluding remarks. 2. New Exponential Inverted Nadarajah–Haghighi Distribution The CDF and PDF of the NEINH are obtained by substituting equations (3) and (4) into (1) and (2) respectively, as follows. F (x) = 1 − 1 − e { 1− ( 1+ κ x )τ} eϑe { 1− ( 1+ κ x )τ} , x > 0, ϑ, τ, κ > 0, (5) f(x) = τκx−2 ( 1 + κ x )τ−1 e { 1− ( 1+ κ x )τ} 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}] e ϑ { e { 1− ( 1+ κ x )τ}} . (6) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 4 of 29 The survival S(x) function is defined as S(x) = 1 − e { 1− ( 1+ κ x )τ} eϑe { 1− ( 1+ κ x )τ} . (7) The hazard rate function (HF), frequently employed in lifetime modelling to represent the probability of failure, is defined as HF = τκx−2 ( 1 + κ x )τ−1 e { 1− ( 1+ κ x )τ} 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}] 1 − e { 1− ( 1+ κ x )τ} . (8) Figure 1: The NEINH density plots. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 5 of 29 Figure 2: The NEINH HF’s plots. Several plots illustrating the PDF and HF of the NEINH model for specific param- eter values are provided. The PDF in Figure (1) displays a variety of shapes, including right-skewed, left-skewed, approximately symmetrical, and decaying forms. Similarly, Figure (2) demonstrates multiple HF patterns, such as inverted, decreasing, and con- stant shapes. These results highlight the substantial flexibility of the NEINH model in adapting to real-world data. 2.1. Linear expression of the NEINH density function A linear representation of NEINH’s PDF has been constructed using mathemati- cal expansions. The derivation begins with the exponential expansion with a negative exponent is given by e−x = ∞∑ i=0 (−1)ixi i! . (9) By applying (9), the NEINH’s PDF is expressed as f(x) = ∞∑ c1=0 (−1)c1ϑc1 c1! τκx−2 ( 1 + κ x )τ−1 e(1+c1) { 1− ( 1+ κ x )τ} { 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}]} . (10) The binomial theorem is given by (1 + x)n = n∑ i=0 ( n i ) xi. (11) By applying (47) twice, the PDF of NEINH is simplified to the following S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 6 of 29 f(x) = η τ κ x−2 ( 1 + κ x )τ−1 e−(1+c1+c3) ( 1+ κ x )τ (12) where η = ∞∑ c1=0 1∑ c2=0 c2∑ c3=0 (−1)(c1+c3)ϑ(c1+c3) c1! ( 1 c2 )( c2 c3 ) e(1+c1+c3) (13) 3. Properties of the NEINH In this section, several characteristic properties of the NEINH are established. 3.1. Quantile Function Let X ∼ NEINH, the uth quantile function (0 < u < 1) can be obtained through inverting the equation (5), and solving the non-linear equation by applying the Lambert function W [.]. xu = κ  1 − log 1 − W ( ϑeϑ(1 − u) ) ϑ   1 τ − 1  −1 , 0 ≤ u ≤ 1. (14) The median of the NEINH is calculated by substituting u = 0.5 into equation (14). Median(x) = κ  1 − log 1 − W ( ϑeϑ(0.5) ) ϑ   1 τ − 1  −1 . (15) 3.2. Moment The rth moment of X ∼ NEINH can be derived as follows: µr = E (xr) = ∫ ∞ 0 xrf(x)dx = η ∫ ∞ 0 τ κ xrx−2 ( 1 + κ x )τ−1 e−(1+c1+c3) ( 1+ κ x )τ dx. (16) By substituting y = ( 1 + κ x )τ , therefore the rth moment is defined as: µr = η ∞∑ c4 (−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, (17) where η is given by (13) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 7 of 29 3.3. Moment Generating Function The moment-generating function (MGF) of the NEINH is expressed as MX(t) = E(etx) = ∞∑ r=0 tr r!µr. (18) Consequently, by substituting (55) into (56), the MGF is derived as MX(t) = ∞∑ r=0 tr r! ∞∑ c4 η (−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, (19) where η is given by (13) 3.4. Characteristic Function The NEINH characteristic function is derived as φx (t) = E(eitx) = ∞∑ r=0 (it)r r! ∞∑ c4 η (−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, (20) where η is given by (13) 3.5. Rényi entropy The Rényi entropy is employed to measure the uncertainty associated with the ran- dom variable X. A higher Rényi entropy value indicates greater uncertainty in the data. According to [35], the Rényi entropy, RE(ϕ), is defined as RE(ϕ) = 1 1 − ϕ log [ ∫ ∞ −∞ [f(x)]ϕdx ] (21) By substituting f(x) given in (6) into the (21) and applying some mathematical expan- sions RE(ϕ) is presented as [f(x)]ϕ = η∗τϕκϕ (−1)c1+c3x−2ϕ ( 1 + κ x )ϕ(τ−1) e−(ϕ+c1+c3) ( 1+ κ x )τ , (22) where η∗ = ∞∑ c1=0 ϕ∑ c2=0 c2∑ c3=0 ( ϕ c2 ) ( c2 c3 ) ϕc1 ϑc1+c2 c1! e(ϕ+c1+c3). (23) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 8 of 29 By replacing (22) in (21), and calculating the integral, the Rényi entropy of the NEINH can be derived as RE(ϕ) = 1 1 − ϕ log [ η∗ ∞∑ c4=0 ( −2ϕ + c4 + 1 c4 ) (−1)c1+c3−2ϕ+2 τϕ−1κ1−ϕ (ϕ + c1 + c3) 1 τ [ϕ(τ−1)−(τ−1)+i]+1 × Γ (1 τ [ϕ(τ − 1) − (τ − 1) + c4] + 1 )] . (24) 3.6. Order statistics Let Xi:n represent the ith order statistic from a random sample X1, X2, ....., Xn of the NEINH distribution. The PDF of the ith order statistic, pi:n(x), is therefore expressed as fi:n(x) = n! (i − 1)!(n − i)!f(x) [F (x)]i−1 [1 − F (x)]n−i . (25) By applying binomial expansion, the PDF of Xi:n become fi:n(x) = n∑ j=0 (−1)jn! (i − 1)!(n − i)! ( n j ) f(x) [F (x)]j+i−1 . (26) Substituting (5) into (26), the PDF of Xi:n fi:n(x) = n∑ j=0 (−1)jn! (i − 1)!(n − i)! ( n j ) f(x) 1 − 1 − e { 1− ( 1+ κ x )τ} eϑe { 1− ( 1+ κ x )τ} j+i−1 , (27) where f(x) given by (6). 4. Estimation of Parameters This section presents seven methods of estimation used to estimate the parameters of the NEINH. 4.1. ML estimation method If x1, ...., xn are NEINH RS of size n, the likelihood function L (Ω), and the log- likelihood function ` (Ω) for Ω = (ϑ, ăτ, κ) are defined as follows: L (Ω) =τn κn n∏ i=1 xi −2 n∏ i=1 ( 1 + κ xi )(τ−1) e {∑n i=1 { 1− ( 1+ κ xi )τ}} n∏ i=1 { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} e− ∑n i=1 ϑe { 1− ( 1+ κ xi )τ} . (28) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 9 of 29 ` (Ω) =n log(τ) + n log(κ) − 2 n∑ i=1 log(xi) + (τ − 1) n∑ i=1 log(1 + κ xi ) + n∑ i=1 { 1 − (1 + κ xi )τ } + n∑ i=1 { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} − n∑ i=1 ϑe { 1− ( 1+ κ xi )τ} . (29) The following are the first derivatives of (29), with respect to Ω = (ϑ, τ, κ) ∂` ∂ϑ = n∑ i=1 [ 1 − e { 1− ( 1+ κ xi )τ}] { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} − n∑ i=1 e { 1− ( 1+ κ xi )τ} , (30) ∂` ∂τ =n τ + n∑ i=1 log ( 1 + κ xi ) − n∑ i=1 ( 1 + κ xi )τ log ( 1 + κ xi ) + n∑ i=1 ϑ ( 1 + κ xi )τ log ( 1 + κ xi ) e { 1− ( 1+ κ xi )τ} { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} + n∑ i=1 ϑ ( 1 + κ xi )τ log ( 1 + κ xi ) e { 1− ( 1+ κ xi )τ} , and (31) ∂` ∂κ =n κ + (τ − 1) n∑ i=1 1 xi ( 1 + κ xi ) − τ n∑ i=1 ( 1 + κ xi )τ−1 + n∑ i=1 ϑτ ( 1 + κ xi )τ−1 e { 1− ( 1+ κ xi )τ} xi { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} + n∑ i=1 ϑτ xi ( 1 + κ xi )τ−1 e { 1− ( 1+ κ xi )τ} . (32) Equations (30-32) can be solved through numerical methods, such as the Newton- Raphson method, which is a commonly used optimization technique. 4.2. Maximum product of spacing estimation method The MPS method, introduced by [36], utilizes the uniform spacings of a random sample from the NEINH distribution with size n to determine the MPS. This method S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 10 of 29 can be applied as follows: Di = F (xi:n) − F (xi−1:n), i = 1, 2, ..., n + 1, (33) where F (i) denotes the CDF of the observation xi:n from the NEINH distribution. The MPS estimates for the NEINH parameters can then be obtained by maximizing the logarithm of the geometric mean of the sample spacings, given by M = 1 n + 1 n+1∑ i=1 log Di. (34) 4.3. Bayesian estimation method In the Bayesian approach, parameters are considered as random variables with pre- defined prior distributions. This method is especially valuable in survival analysis due to its capacity to incorporate prior knowledge into the research. One of the primary challenges in survival analysis is the scarcity of available data. For ϑ, τ , and κ are selected as gamma distributions. Hence, π1(ϑ) ∝ ϑd11−1e−d12ϑ, ϑ > 0, d11, d12 > 0, π2(τ) ∝ τd21−1e−d22τ , τ > 0, d21, b22 > 0, π3(κ) ∝ κd31−1e−d32κ, κ > 0, d31, d32 > 0. (35) Assuming that the parameters of the proposed model are independent, The joint distri- bution of the priors is obtained as follows: π(ϑ, τ, κ) ∝ ϑd11−1τd21−1κd31−1e−(d12ϑ+d22τ+d32κ). (36) The posterior function of the parameters of the proposed distribution can be computed using equations (28) and (36) as follows. π∗(Ω|X) ∝ L(ϑ, τ, κ)π(ϑ, τ, κ) ∝ ϑd11−1τn+d21−1 κn+d31−1 n∏ i=1 xi −2 n∏ i=1 ( 1 + κ xi )(τ−1) × e−(d22τ+d32κ) e {∑n i=1 { 1− ( 1+ κ xi )τ}} × n∏ i=1 { 1 + ϑ [ 1 − e { 1− ( 1+ κ xi )τ}]} e −ϑ { d12+ ∑n i=1 e { 1− ( 1+ κ xi )τ}} . (37) Bayesian parameter estimation for the NEINH distribution is calculated by ϑ̂ = ∫ ∞ 0 ϑ ∫ ∞ 0 ∫ ∞ 0 π∗(ϑ|x) dτdκdϑ (38) τ̂ = ∫ ∞ 0 τ ∫ ∞ 0 ∫ ∞ 0 π∗(ϑ|x) dκdϑdτ (39) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 11 of 29 κ̂ = ∫ ∞ 0 κ ∫ ∞ 0 ∫ ∞ 0 π∗(ϑ|x) dϑdτdκ (40) The integrals in equations (38), (39), and (40) are particularly complex. The Metropolis- Hastings (MH) algorithm is used to derive approximations for these integrals. 4.4. Ordinary and weighted least squares estimation methods The OLS and WLS are introduced by [37]. Suppose that x1:n ≤ ... ≤ xi:n ≤ ... ≤ xn:n represent the order statistics of a RS from the NEINH. The OLS can be found by minimizing a sum of squared differences between the theoretical and the empirical CDF functions with respect to the parameters of NEINH as O = n∑ i=1 1 − 1 − e { 1− ( 1+ κ xi )τ} eϑe { 1− ( 1+ κ xi )τ} − F(i)  2 , (41) where F(i) represents the empirical CDF of the observation xi:n of the NEINH which is usually estimated by F(i) = i/(n + 1). The WLS estimates are obtained similarly to the OLS estimates, but with a weighted sum of squared differences as follows. W = n∑ i=1 wi 1 − 1 − e { 1− ( 1+ κ xi )τ} eϑe { 1− ( 1+ κ xi )τ} − F(i)  2 , wi = (n + 1)2(n + 2) i(n + 1 − i) . (42) 4.5. Cramér von Mises and Anderson Darling estimation methods The CM and AD estimate methods are presented by [38] in the context of statistical testing. Their foundation comes from the difference between the CDF estimates and the empirical distribution function. The functions (43) and (44) minimized with respect to the parameters of NEINH to find CM and AD estimates, respectively C = 1 12n + n∑ i=1 1 − 1 − exp { 1 − ( 1 + κ xi )τ} eϑe { 1− ( 1+ κ xi )τ} − 2i − 1 2n 2 , (43) A = −n − 1 n n∑ i=1 (2i − 1) [ log F (xi:n) + log F (xn−i+1:n) ] . (44) 5. Simulation Studies In this section, the effectiveness of various estimation methods is assessed through numerical simulations. From the NEINH distribution, a total of N = 1000 samples was randomly generated, using two sets of parameter values, with sample sizes of n = 30, 100, 200, and 500. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 12 of 29 • SetI : ϑ = 1.6, τ = 0.5, κ = 0.1 • SetII : ϑ = 6, τ = 0.25, κ = 1.7 Monte Carlo simulation was used to obtain NEINH  parameter estimates for the seven methods. All estimation results were obtained using the R programming. For each parameter, we calculated the mean estimates and the mean square error (MSE), defined as MSE = var(Ω̂) + [Bias(Ω̂)]2 = 1 N n∑ i=1 (Ω̂−Ωtr)2, where, Ω = (ϑ, τ, κ) and Bias = 1 N ∑n i=1 ( Ω̂ − Ωtr ) . The ML, MPS, Bayesian, OLS, WLS, CM, and AD parameter estimates were ob- tained along with their MSE as shown in Tables (1)-(2). According to Tables (1)-(2), for most methods, the MSE decreases as the sample size increases. Additionally, parameter estimates tend to converge to true values as sample size increases. In terms of MSE, Bayesian estimation proves to be the most effective method for estimating the NEINH parameter, outperforming all other estimation tech- niques. Following Bayesian estimation, the ML and WLS methods also demonstrate strong performance. In contrast, the OLS, CM and AD estimators yield similar results, but tend to produce larger MSE values, particularly when the sample size is small. How- ever, as the sample size increases to 200 or 500, the accuracy of all estimators converges, with their performance becoming approximately equivalent. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 13 of 29 Table 1: NEINH parameters estimators and MSE for Set I. n ML MPS Bayesian OLS WLS CM AD ϑ̂ 1.6516 2.8819 1.5794 3.4492 1.9248 3.7545 3.9278 (4.53 e-01) (5.7709) (1.95 e-02) (2.83 e+01) (4.94 e-01) (3.17 e+01) (5.89 e+01) 30 τ̂ 0.5822 0.4420 0.5203 0.5061 0.5467 0.5308 0.5382 (3.64 e-02) (1.81 e-02) (4.31 e-03) (3.78 e-02) (3.25 e-02) (4.43 e-02) (5.22 e-02) κ̂ 0.1079 0.4663 0.0997 1.0927 0.1365 0.9591 1.1239 (1.02 e-02) (8.77 e-01) (5.26 e-04) (2.36 e+01) (1.84 e-02) (1.94 e+01) (1.97 e+01) ϑ̂ 1.710 2.2565 1.5750 2.2912 1.7018 2.3915 2.0281 (4.06 e-01) (2.5933) (1.85 e-02) (5.3448) (1.64 e-01) (5.7669) (2.9659) 100 τ̂ 0.5212 0.4648 0.5108 0.5081 0.5169 0.5135 0.5231 (8.30 e-03) (7.81 e-03) (2.13 e-03) (1.77 e-02) (7.81 e-03) (1.90 e-02) (1.86 e-02) κ̂ 0.1183 0.2613 0.0999 0.3070 0.1092 0.3085 0.2108 (8.57 e-03) (2.60 e-01) (4.22 e-04) (7.20 e-01) (3.08 e-03) (7.21 e-01) (2.47 e-01) ϑ̂ 1.7299 1.9743 1.5923 1.9141 1.7046 1.9585 1.7829 (3.51 e-01) (1.1792) (1.66 e-02) (1.5940) (1.40 e-01) (1.6763) (9.37 e-01) 200 τ̂ 0.5083 0.4798 0.5054 0.5046 0.5049 0.5064 0.5143 (5.53 e-03) (5.64 e-03) (1.29 e-03) (9.85 e-03) (4.16 e-03) (9.99 e-03)) (9.75 e-03) κ̂ 0.11701 0.1652 0.1012 0.1634 0.1104 0.1646 0.1334 (4.60 e-03) (4.23 e-02) (2.99 e-04) (5.50 e-02) (1.99 e-03) (5.57 e-02) (1.62 e-02) ϑ̂ 1.6565 1.6959 1.5936 1.7333 1.6507 1.7497 1.6705 (1.39 e-01) (2.12 e-01) (1.51 e-02) (4.64 e-01) (7.42 e-02) (4.75 e-01) (3.39 e-01) 500 τ̂ 0.5034 0.4945 0.5031 0.5004 0.5028 0.5010 0.5051 (2.27 e-03) (1.26 e-03) (4.98 e-04) (4.65 e-03) (1.91 e-03) (4.67 e-03) (3.98 e-03) κ̂ 0.1066 0.1139 0.1002 0.1226 0.1048 0.1230 0.1131 (1.34 e-03) (5.36 e-03) (1.55 e-04) (6.11 e-03) (8.02 e-04) (6.18 e-03) (3.30 e-03) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 14 of 29 Table 2: NEINH parameters estimators and MSE for Set II. n ML MPS Bayesian OLS WLS CM AD ϑ̂ 5.9811 6.5403 5.9997 6.2947 6.4993 6.3630 5.7915 (1.8997) (4.4987) (8.51 e-05) (5.0136) (2.9104) (5.4772) (3.1845) 30 τ̂ 0.2649 0.2500 0.2509 0.2505 0.2678 0.2603 0.2607 (2.18 e-03) (2.38 e-03) (1.07 e-04) (1.52 e-03) (2.62 e-03) (1.05 e-02) (2.45 e-03) κ̂ 1.7104 2.3398 1.6995 2.4451 1.7818 2.2771 1.8991 (1.0201) (3.4656) (2.61 e-04) (4.7756) (1.4845) (4.0528) (1.5404) ϑ̂ 5.9993 6.4250 5.9995 5.9542 6.2683 6.0333 5.8606 (8.78 e-01) (3.0018) (7.61 e-05) (2.2520) (1.0812) (2.1904) (1.5670) 100 τ̂ 0.2544 0.2498 0.2500 0.2546 0.2550 0.2561 0.2537 (4.14 e-04) (6.63 e-04) (2.97 e-05) (9.94 e-04) (5.61 e-04) (9.04 e-04) (5.94 e-04) κ̂ 1.7298 2.1397 1.6995 1.895 1.8213 1.8647 1.8253 ( 4.77 e-01) (1.8198) (2.25 e-04) (1.4906) (7.14 e-01) (1.4513) (9.16 e-01) ϑ̂ 6.0371 6.3856 6.0001 5.9328 6.1883 5.9321 5.9153 (1.46 e-01) (2.0893) (6.39 e-05) (1.1857) (6.02 e-01) (1.2802) (1.0587) 200 τ̂ 0.2513 0.2475 0.2502 0.2531 0.2531 0.2546 0.2520 (1.91 e-04) (3.33 e-04) (1.31 e-05) (3.74 e-04) (2.88 e-04) (4.67 e-04) (3.32 e-04) κ̂ 1.765 2.1679 1.6996 1.7723 1.7714 1.7408 1.7957 (3.18 e-01) (1.8458) (2.12 e-04) (6.87 e-01) (3.97 e-01) (6.53 e-01) (5.88 e-01) ϑ̂ 6.0117 6.3653 5.9999 5.9710 6.1079 5.9857 5.9927 (3.05 e-01) (1.4392) (6.31 e-05) (7.43 e-01) (3.19 e-01) (7.65 e-01) (6.30 e-01) 500 τ̂ 0.2500 0.2474 0.2499 0.2514 0.2505 0.2517 0.2503 (9.08 e-05) (1.66 e-04) (4.59 e-06) (2.05 e-04) (1.09 e-04) (1.88 e-04) (1.60 e-04) κ̂ 1.7003 2.0776 1.7004 1.7536 1.7701 1.7417 1.8009 (1.78 e-01) (1.2500) (1.96 e-04) (3.27 e-01) (2.04 e-01) (3.16 e-01) (3.85 e-01) 6. Applications This section illustrates the practical utility of the NEINH distribution by applying it to four real-world medical datasets. The datasets are given below. To assess the adequacy of the NEINH distribution, its performance is compared against four competing distributions, with their CDFs defined as follows: • The inverted Nadarajah Haghighi (INH) distribution given in (3) • Power Inverted Nadarajah–Haghighi (PINH) Distribution [32] F (x) = e { 1− ( 1+ κ xγ )τ} ; x > 0, τ, κ, γ > 0. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 15 of 29 • Extended Odd Weibull Inverse Nadarajah Haghighi (EOWINH) Distribution [31] F (x) = 1 − 1 + c [ e { 1− ( 1+ κ x )τ} 1−e { 1− ( 1+ κ x )τ}]d  −1 c ; x > 0, τ, κ, c, d > 0. • Half Logistic Inverted Nadarajah Haghighi (HLINH) Distribution [34] F (x) = 1− [ 1−e { 1− ( 1+ κ x )τ}]v 1+ [ 1−e { 1− ( 1+ κ x )τ}]v ; x > 0, τ, κ, v > 0 • The odd Lomax inverted Nadarajah Haghighi (OLINH) distribution [33] F (x) = 1 − ba [ b + e { 1− ( 1+ κ x )τ} 1−e { 1− ( 1+ κ x )τ}]−a ; x > 0, τ, κ, a, b > 0. • the Nadarajah–Haghighi (NH) distribution [39] F (x) = 1 − e{1−(1+κx)τ }; x > 0, τ, κ > 0. Data 1: Blood cancer This data provides the survival duration (in years) for 40 leukemia patients treated at a Ministry of Health hospital in Saudi Arabia, as reported by [40]. 0.315 0.496 0.616 1.145 1.208 1.263 1.414 2.025 2.036 2.162 2.211 2.370 2.532 2.693 2.805 2.910 2.912 3.192 3.263 3.348 3.348 3.427 3.499 3.534 3.767 3.751 3.858 3.986 4.049 4.244 4.323 4.381 4.392 4.397 4.647 4.753 4.929 4.973 5.074 5.381 S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 16 of 29 Table 3: Analysis of MLs and GoFs for data 1 Distributions NEINH INH PINH HLINH OLINH NH Estimates ϑ̂ = 4.12e+04 τ̂ = 0.2418 τ̂ = 0.9907 τ̂ = 18.8841 τ̂ = 0.5688 τ̂ = 0.3483 τ̂ = 5.4006 κ̂ = 8.85e+04 κ̂ = 2.0420 κ̂ = 0.0537 κ̂ = 18.0788 κ̂ = 541.9925 κ̂ =0.0451 γ̂ = 0.7218 v̂ = 9.0194 â =189.8362 b̂ =1.4510 −` -74.0331 -91.4842 -97.3297 -74.6817 -74.2407 -77.9047 CAIC 154.7330 187.2928 201.3261 158.5064 157.6244 160.1338 AIC 154.0663 186.9685 0 200.6595 157.3636 156.4815 159.8095 BIC 159.1329 190.3462 205.7261 164.1191 163.2370 163.1872 HQIC 155.8982 188.1898 202.4914 159.8062 158.9241 161.0308 K-S 0.1178 0.3129 0.2942 0.1674 0.1648 0.2786 p-value 0.6345 0.0007 0.0019 0.2119 0.2268 0.0040 Figure 3: The NEINH distribution is evaluated against other distributions using data 1. (Right): CDFs of the distributions. (Left): observed versus expected frequencies for each distribution. Data 2: Head and Neck cancer The survival data of 44 patients diagnosed with head and neck cancer are examined. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 17 of 29 The data were originally reported by [41] and subsequently analyzed by [42]. 12.20 23.56 23.74 25.87 31.98 37 41.35 47.38 55.46 58.36 63.47 68.46 78.26 74.47 81.43 84 92 94 110 112 119 127 130 133 140 146 155 159 173 179 194 195 209 249 281 319 339 432 469 519 633 725 817 1776 Table 4: Analysis of MLs and GoFs for data 2 Distributions NEINH INH PINH HLINH OLINH NH Estimates ϑ̂ = 5.0284 τ̂ = 0.3737 τ̂ = 0.8516 τ̂ = 0.4465 τ̂ = 0.7329 τ̂ = 155.4218 τ̂ = 0.6877 κ̂ = 2599.9991 κ̂ = 99.7859 κ̂ = 2103.9957 κ̂ = 129.6323 κ̂ = 0.1205 κ̂ =0.0086 γ̂ = 1.4338 v̂ = 1.6545 â =2.4432 b̂ =16.7024 −` -277.4739 279.3727 -277.7396 -277.5251 -277.5157 -280.8588 CAIC 561.5477 563.0380 562.0793 561.6501 564.0570 566.0103 AIC 560.9477 562.7453 561.4793 561.0501 563.0314 565.7177 BIC 566.3003 566.3137 566.8319 566.4027 570.1681 569.2860 HQIC 562.9327 564.0686 563.4643 563.0351 565.6780 567.0410 K-S 0.0552 0.0758 0.0646 0.0645 0.0627 0.1051 p-value 0.9982 0.9450 0.9870 0.9873 0.9907 0.6765 S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 18 of 29 Figure 4: The NEINH distribution is evaluated against other distributions using data 2. (Right): CDFs of the distributions. (Left): observed versus expected frequencies for each distribution. Data 3: Acute bone cancer The survival times, measured in days, of 73 patients diagnosed with acute bone cancer are provided in, [43]: 0.09 0.76 1.81 1.10 3.72 0.72 2.49 1.00 0.53 0.66 31.61 0.60 0.20 1.61 1.88 0.70 1.36 0.43 3.16 1.57 4.93 11.07 1.63 1.39 4.54 3.12 86.01 1.92 0.92 4.04 1.16 2.26 0.20 0.94 1.82 3.99 1.46 2.75 1.38 2.76 1.86 2.68 1.76 0.67 1.29 1.56 2.83 0.71 1.48 2.41 0.66 0.65 2.36 1.29 13.75 0.67 3.70 0.76 3.63 0.68 2.65 0.95 2.30 2.57 0.61 3.93 1.56 1.29 9.94 1.67 1.42 4.18 1.37 S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 19 of 29 Table 5: Analysis of MLs and GoFs for data 3 Distributions NEINH INH PINH HLINH OLINH NH Estimates ϑ̂ = 3.9488 τ̂ = 0.4440 τ̂ = 0.7596 τ̂ = 0.3254 τ̂ = 0.6073 τ̂ = 0.4302 τ̂ = 0.5223 κ̂ = 16.7693 κ̂ = 1.6226 κ̂ = 9.2579 κ̂ = 2.9132 κ̂ = 24.8730 κ̂ =1.04362 γ̂ = 1.8333 v̂ = 2.0633 â = 1.7242 b̂ =0.2033 −` -141.0170 -146.8324 -142.1592 -142.2841 -141.2973 -154.0040 CAIC 288.3818 297.8363 290.6663 290.9160 291.1828 312.1795 AIC 288.0340 297.6649 290.3184 290.5682 290.5946 312.0080 BIC 294.9053 302.2458 297.1898 297.4396 299.7564 316.5889 HQIC 290.7723 299.4904 293.0568 293.3065 294.2457 313.8336 K-S 0.0880 0.1505 0.0916 0.0926 0.0926 0.1825 p-value 0.6225 0.0730 0.5718 0.5571 0.5575 0.0153 Figure 5: The NEINH distribution is evaluated against other distributions using data 3. (Right): CDFs of the distributions. (Left): observed versus expected frequencies for each distribution. Data 4: Pain Relief Times This dataset contains the pain relief times, recorded in minutes, for 20 patients who received an analgesic. Originally reported by [44], the data has also been reanalyzed by [45] and [46]. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 20 of 29 1.1 1.4 1.3 1.7 1.9 1.8 1.6 2.2 1.7 2.7 4.1 1.8 1.5 1.2 1.4 3.0 1.7 2.3 1.6 2.0 Table 6: Analysis of MLs and GoFs for data 4 Distributions NEINH INH PINH HLINH OLINH NH Estimates ϑ̂ = 6.7462 τ̂ = 21.9181 τ̂ = 88.9201 τ̂ =0.3344 τ̂ = 36.5257 τ̂ = 0.4381 τ̂ = 55.0732 κ̂ = 0.1001 κ̂ = 0.0148 κ̂ = 162.9943 κ̂ = 0.0521 κ̂ = 231.2687 κ̂ =0.0069 γ̂ = 6.4866 v̂ = 6.6494 â = 4.910 b̂ =0.0038 −` -15.5056 -26.7583 -16.1470 -15.9439 -17.6737 -27.8239 CAIC 38.5113 58.2224 39.79404 39.3879 46.0140 60.3537 AIC 37.0113 57.5166 38.2940 37.8879 43.3474 59.6478 BIC 39.9985 59.5080 41.2812 40.8750 47.3303 61.6393 HQIC 37.5944 57.9053 38.8771 38.4710 44.1249 60.0365 K-S 0.0983 0.3866 0.1362 0.1292 0.1462 0.4054 p-value 0.9903 0.0050 0.8519 0.8918 0.7860 0.0027 S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 21 of 29 Figure 6: The NEINH distribution is evaluated against other distributions using the data 4. (Right): CDFs of the distributions. (Left): observed versus expected frequencies for each distribution. The parameters for each distribution were estimated using ML, and their respective log-likelihoods were obtained. To evaluate the NEINH distribution’s performance, sev- eral goodness-of-fit (GoF) tests were applied, such as the corrected Akaike information criterion (CAIC), Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC), and the Kolmogorov-Smirnov (K- S) test. The K-S test also provided the corresponding p-value. The optimal model is identified by the lowest values for these criteria and the highest p-value. MLEs of the parameters, along with the log-likelihood values and GoF statistics for each model, are shown in Tables (3-6). These results suggest that the NEINH distri- bution consistently provides the lowest values for CAIC, AIC, BIC, HQIC, and K-S statistics while delivering the most favorable p-values when compared to the other mod- els. Furthermore, Figures (3-6) illustrate that the NEINH distribution closely fits the observed data. Thus, compared to other models, NEINH is the best-fitting distribution for the datasets under analysis. 7. Concluding Remarks The Exponential Inverted Nadarajah–Haghighi (NEINH) distribution, derived from the NLTE-X family, is introduced in this article to offer enhanced flexibility in analyzing real-world data. Its adaptable hazard rate function allows it to model a variety of hazard S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 22 of 29 behaviours, making it applicable in fields like medicine, biology, and engineering. To estimate its parameters, seven methods ML, MPS, Bayesian, OLS, WLS, CM, and AD were employed. A comprehensive simulation study revealed that Bayesian performed the best, as indicated by the lowest MSEs. Additionally, the NEINH distribution was tested on four medical datasets, where it provided a better fit compared to competing models. 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Journal of Nonlinear Science and Applica- S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 25 of 29 tions, 13(5):223–238, 2020. [44] Alan J Gross and Virginia Clark. Survival distributions: reliability applications in the biomedical sciences. John Wiley, New York., 1975. [45] Rama Shanker, F Hagos, and S Sujatha. On modeling of lifetime data using one parameter akash, lindley and exponential distributions. Biometrics & Biostatistics International Journal, 3(2):1–10, 2016. [46] Rana Ali Bakoban and Ashwaq Mohammad Al-Shehri. A new generalization of the generalized inverse rayleigh distribution with applications. Symmetry, 13(4):711, 2021. Appendix A. Proof (1) We need to find the linear representation of NEINH’s PDF which is presented in (6). The derivation begins with the exponential expansion with a negative exponent is given in (9) By applying (9) into term e ϑ { e { 1− ( 1+ κ x )τ}} in (6), the NEINH’s PDF is expressed as f(x) = ∞∑ c1=0 (−1)c1ϑc1 c1! τκx−2 ( 1 + κ x )τ−1 e(1+c1) { 1− ( 1+ κ x )τ} { 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}]} . (45) By applying (11) into term { 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}]} in (45), we got the following. { 1 + ϑ [ 1 − e { 1− ( 1+ κ x )τ}]} = 1∑ c2=0 ( 1 c2 ) ϑc2 [ 1 − e { 1− ( 1+ κ x )τ}]c2 (46) Again, the following binomial theorem is used. (1 − x)n = n∑ i=0 ( n i ) (−1)ixi. (47) By applying (47) into term [ 1 − e { 1− ( 1+ κ x )τ}] in (46), we get the following.[ 1 − e { 1− ( 1+ κ x )τ}]c2 = c2∑ c3=0 ( c2 c3 ) (−1)c3 [ e { 1− ( 1+ κ x )τ}]c3 (48) After the simplification in (46) and (48), the PDF of NEINH is written as the fol- lowing. f(x) = ∞∑ c1=0 1∑ c2=0 c2∑ c3=0 (−1)(c1+c3)ϑ(c1+c3)τκ c1! ( 1 c2 )( c2 c3 ) e(1+c1+c3)x−2 ( 1 + κ x )τ−1 e−(1+c1+c3) ( 1+ κ x )τ , (49) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 26 of 29 We can put the constant in one symbol (η). , the PDF of NEINH is simplified as follows. f(x) = ητκx−2 ( 1 + κ x )τ−1 e−(1+c1+c3) ( 1+ κ x )τ , where η = ∞∑ c1=0 1∑ c2=0 c2∑ c3=0 (−1)(c1+c3)ϑ(c1+c3) c1! ( 1 c2 )( c2 c3 ) e(1+c1+c3) B. Proof (2) The rth moment of X ∼ NEINH can be derived as follows: µr = E (xr) = ∫ ∞ 0 xrf(x)dx (50) = η ∫ ∞ 0 τ κxrx−2 ( 1 + κ x )τ−1 e−(1+c1+c3) ( 1+ κ x )τ dx. (51) Substitute x in (51) by y = ( 1 + κ x )τ as follows. y = ( 1 + κ x )τ dy = τ ( 1 + κ x )τ−1 κx−2dx y1/τ = 1 + κ x dx = dyx2 τκ ( 1 + κ x )τ−1 x = −κ ( 1 − y1/τ )−1 Therfore, µr = η ∫ ∞ 0 (−1)rκr ( 1 − y1/τ )−r e−(1+c1+c3)ydy. (52) By using the binomial theorem (53) into term ( 1 − y1/τ )−r in (52) as follows. (1 − x)−n = ∞∑ i=0 ( n + 1 − i i ) xi. (53) (1 − y1/τ )−r = ∞∑ c4=0 ( r + 1 − c4 i ) y c4 τ . (54) Therfore, µr = η(−1)rκr ∞∑ c4=0 ( r + 1 − c4 i ) 1 (1 + c1 + c3) c4 τ +1 ∫ ∞ 0 [(1 + c1 + c3)y] c4 τ +1−1 e−(1+c1+c3)y(1 + c1 + c3)dy. Therefore, by using the gamma function, the µr is written as follows. S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 27 of 29 µr = η ∞∑ c4 (−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, (55) where η is given by (13). The moment-generating function (MGF) and the characteristic function of the NEINH are derived by substituting the µr in (56) and (57) by (55). MX(t) = E(etx) = ∞∑ r=0 tr r!µr. (56) φx (t) = E(eitx) = ∞∑ r=0 (it)r r! µr. (57) Therefore, the moment-generating function, MX(t) and φx (t) of the NEINH are written as follows. MX(t) = ∞∑ r=0 tr r! ∞∑ c4 η(−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, φx (t) = ∞∑ r=0 (it)r r! ∞∑ c4 η(−1)rκr ( r + c4 − 1 c4 ) Γ ( c4 τ + 1 ) (1 + c1 + c3) c4 τ +1 , c4 τ + 1 > 0, where η is given by (13) C. R Code # NEINH # PDF dNEINH=function (x , pars ){ th=pars [ 1 ] ; a=pars [ 2 ] ; b=pars [ 3 ] p1=(1+(b/x ))^ a p11=(1+(b/x ) )^ ( a−1) p2=exp(1−p1 ) p3=1−p2 p4=exp( th∗p2 ) p5=1+(th∗p3 ) S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 28 of 29 pdf=a∗b∗x^(−2)∗p11∗p2∗(1/p4 )∗p5 return ( pdf ) } # CDF pNEINH=function (q , th , a , b ){ p1=(1+(b/q))^ a p2=exp(1−p1 ) p5=exp( th∗p2 ) cd f=1−((1−p2 )/p5 ) return ( cd f ) } # Quant i l e qNEINH=function (p , pars ){ th=pars [ 1 ] ; a=pars [ 2 ] ; b=pars [ 3 ] u=runif (p , 0 , 1 ) z=th∗exp( th )∗(1−u) y=log (1−(lambertW0 ( z )/th ) ) f=(1−y )^(1/a ) k=(f −1)^(−1) x=b∗k return ( x )} # hazard ra t e func t i on h ( x ) hNEINH=function (x , th , a , b){#qEIW h=dNEINH(x , th , a , b )/(1−pNEINH(x , th , a , b ) ) return (h) } # l o g l i k e l i h o o d func t i on lh_NEINH=function (par , x ){ L=sum( log (dNEINH(x , par [ 1 ] , par [ 2 ] , par [ 3 ] ) ) ) return(−L)} ##################################################################### #In t h i s program Bayesian methods are used to e s t ima t ion the parameters # ##################################################################### ## Define the Prior d i s t r i b u t i o n s : p r i o r_MO=function ( pars ) { th=pars [ 1 ] ; a=pars [ 2 ] ; b=pars [ 3 ] p r i o r_th=dgamma( th , 5 , 0 . 2 0 , log=TRUE)#5 p r i o r_a=dgamma( a , 5 , 0 . 2 0 , log=TRUE)#5 p r i o r_b=dgamma(b , 5 , 0 . 2 0 , log=TRUE)#5 return ( p r i o r_th + p r i o r_a +p r i o r_b) } S. F. Aloufi, I. A. Alsaggaf / Eur. J. Pure Appl. Math, 18 (4) (2025), 6121 29 of 29 ## Dfined the f u n c t i o n s ############################## ##1: The l i k e l i h o o d func t i on o f f o r Bayesian method Bayes . l i k l=function ( pars , data ) { x=data p r i o r= p r i o r_MO( pars ) log_l i k e l i h o o d=sum( log (dNEINH(x , pars ) ) ) return ( log_l i k e l i h o o d + p r i o r ) } ############################################################## ############################################################## Estimate0=array ( c ( 0 ) ,dim=c ( 3 , 3 , 4 ) ,dimnames=l i s t ( c ( ” theta ” , ” alpha ” , ” beta ” ) , c ( ” Estimate ” , ” Bias ” , ”MSE” ) , c ( ”n=30” , ”n=100” , ”n=200” , ”n=500” ) ) )# n=30 ## sample s i z e m=1000## number o f i t e r a t i o n conv=c ( ) ; mess=c ( ) thB0=c ( ) ; aB0=c ( ) ; bB0=c ( ) ############################################################# param=c ( 1 . 6 , 0 . 5 , 0 . 1 ) set . seed (123456789) sample=qNEINH(n∗m, param ) x= matrix (sample , nrow = n , ncol=m) for ( i in 1 :m){ x2=x [ , i ] mcmc_r=Metro_Hast ings ( l i_func=Bayes . l i k l , pars=c ( 1 . 6 , 0 . 5 , 0 . 1 ) , par_names=c ( ” theta ” , ” alpha ” , ” beta ” ) , data=x2 ) thB0 [ i ]=mean(mcmc_r$trace [ , 1 ] ) aB0 [ i ]=mean(mcmc_r$trace [ , 2 ] ) bB0 [ i ]=mean(mcmc_r$trace [ , 3 ] ) print ( i )} Introduction New Exponential Inverted Nadarajah–Haghighi Distribution Linear expression of the NEINH density function Properties of the NEINH Quantile Function Moment Moment Generating Function Characteristic Function Rényi entropy Order statistics Estimation of Parameters ML estimation method Maximum product of spacing estimation method Bayesian estimation method Ordinary and weighted least squares estimation methods Cramér von Mises and Anderson Darling estimation methods Simulation Studies Applications Concluding Remarks Appendix Proof (1) Proof (2) R Code