EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 5860 ISSN 1307-5543 – ejpam.com Published by New York Business Global Statistical Inference on Type II Topp Leone Generalized Inverted Exponential Distribution with Some Applications on Real Data Zakeia A. Al-saiary1,∗, Sarah H. Al-jadaani1 1 Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 22254, Saudi Arabia Abstract. In this article, different Bayes techniques are applied to derive the estimators of the additional shape parameter (α) of Type II Topp Leone Generalized Inverted Exponential (TI- ITLGIE) distribution in the case of complete samples. Two cases were adopted: (a) Bayesian estimation with non-informative priors (Jeffrey’s prior and Quasi prior under three different risk functions: Relative Quadratic loss function (Rq), Precautionary loss function (P) and Entropy loss function (E)). (b) Bayesian estimation with informative prior depending on standard Bayesian technique is derived under three types of loss functions: Squared Error loss function (SE), Linear Exponential loss function (LINEX), and General Entropy loss function (GE). Mont Carlo sim- ulation study was conducted to find the estimators and compare between these techniques and the Non-Bayesian techniques. Some results are given and showed the best estimation method. Numerical computations and real data sets are given to illustrate these results. 2020 Mathematics Subject Classifications: 62F15, 62C10, 62P20 Key Words and Phrases: Bayesian estimation, generalized inverted exponential distribution, Type II Topp–Leone, loss function, Jeffrey’s prior and Quasi prior, Mont Carlo simulation 1. Introduction Some statistical models are still of interest to statisticians. One from these models was Generalized Inverted Exponential (GIE) distribution witch proposed by [1]. It is a flexible model because it has various shapes of the hazard function. There are many gen- eralizations of GIE distribution has discussed by a statisticians, see [2],[3]. Type II Topp Leone generalized inverted exponential distribution is a generalization of GIE distribution witch deduced by [4]. They derived the MLE of its parameters. They proved that the TIITLGIE distribution is more flexible than the baseline GIE distribution. There are some researchers studied the estimation parameters of GIED. [5] used different methods ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.5860 Email addresses: zaalseare@uj.edu.sa (Z. Al-saiary), 2000277@uj.edu.sa (S. Al-jadaani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 2 of 27 of estimation to estimate the unknown parameters of GIE such as MLE, least square and weighted least square methods. Else, [6] derived Bayesian estimators for the shape parameter of GIE distribution using normal, Lindley’s, and Tierney and Kadane’s ap- proximation methods. They applied the Monte Carlo simulations for comparison between various estimates. Moreover, [7] estimated the maximum likelihood and Bayes methods of generalized inverted exponential distribution. [8] obtained the MLEs and asymptotic confidence intervals for the unknown parameters of the generalized inverted exponential distribution. On the other hand, [9] proposed a new continuous distribution called Topp Leone distribution, this distribution is attractive as a generator. Also, many authors were interested in this distribution. The TL distribution had not received much attention un- til by [10] discovered it. The Topp Leone (TL-G) family was proposed by [11]. They introduced the general expression for density and distribution functions for this family. Furthermore, [12] deduced The power inverted Topp Leone power series (PITLPS) class is a novel three-parameter version of the power inverted Topp Leone (PITL) distribution. This compounding process allows for the creation of flexible classes of distributions with strong physical interpretations in fields such as biology and engineering. [13] proposed Topp-Leone Kumaraswamy Marshall-Olkin. They obtained the Shannon entropy, order statistics, the quantile power series, and several associated measures and functions. In this article we will find the Bayesian estimators of the shape parameter of the TIITL- GIE distribution using different techniques. We will compare between those techniques and the MLE using Monte Carlo simulations and Mathematica program depends on the mean square error and the bias. Finally, verify these results using real data sets from different fields. The cumulative distribution function (CDF) and probability density function (PDF) of TIITLGIE distribution with three parameters (α,λ,θ) is introduced by [4] F (x) = 1− [1− [1− [1− e −λ x ]θ]2]α, x > 0, λ, θ, α > 0, (1) and f(x) = 2αθλ x2 e −λ x [1− e −λ x ]θ−1[1− [1− e −λ x ]θ][1− [1− [1− e −λ x ]θ]2]α−1, x > 0, λ, θ, α > 0, (2) where, λ is scale parameter and θ, α are shape parameters. The quantile function of random variable X of TIITL-GIED is given by: x = Q(u) = −λ log[1− [1− [1− [1− u] 1 α ] 1 2 ] 1 θ ] , 0 < u < 1. (3) Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 3 of 27 2. Maximum Likelihood Estimator for shape parameter α Let x1, x2, ..., xn be a random sample of size n from TIITLGIE distribution the likelihood function L(x, α, θ, λ) is written as follows: L(x) = (2αθλ)n n∏ i=1 1 x2i e ∑n i=1 −λ xi n∏ i=1 (1− e −λ xi )θ−1 n∏ i=1 [1− (1− e −λ xi )θ] × n∏ i=1 [1− [1− (1− e −λ xi )θ]2]α−1. (4) Therefore, the log-likelihood function is given as follows log(L) = nlog(2) + nlog(α) + nlog(θ) + nlog(λ)− 2 n∑ i=1 log(xi)− n∑ i=1 λ xi + (θ − 1) × n∑ i=1 log(1− e −λ xi ) + n∑ i=1 log(1− (1− e −λ xi )θ) + (α− 1) n∑ i=1 log(1− (1− (1− e −λ xi )θ)2). Thus on solving ∂ ∂α(log(L)) = 0, we get the ML estimator of shape parameter as α̂ = [− 1 n n∑ i=1 log(1− (1− (1− e −λ̂ xi )θ̂)2)]−1 (5) 3. Bayesian Estimators of the TIITL-GIE distribution for shape parameter α In this section, Bayesian techniques are discussed for estimating the shape parameter α of TIITLGIE distribution under the assumption that the both parameters λ and θ are known based on the complete samples. These estimators are derived using two cases: In the first case we used Bayesian estimation with non-informative priors (Jeffrey’s prior and Quasi prior under three different risk functions: Relative Quadratic loss function (Rq), Precautionary loss function (P) and Entropy loss function (E)). In the second case, we derive the standard Bayesian estimators with Gamma prior under three types of loss functions: squared error loss function, linear exponential loss function and general entropy loss function. 3.1. Case 1: Bayesian estimation with non-informative priors (i) Posterior Distribution under Jeffrey’s prior Let x1, x2, ..., xn be a random sample of size n having TIITL-GIED, the Jeffrey’s Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 4 of 27 non-informative prior for parameter α is given by g1(α) ∝ 1 α , α > 0. (6) By using the Bayes theorem, we get π1(α|x) ∝ L(x|α)g1(α). (7) By substituting Equation (4) and Equation (6) in Equation (7), we have π1(α|x) = A1α n−1exp[−α n∑ i=1 log[1− [1− (1− e −λ xi )θ]2]−1]. π1(α|x) = A1α n−1e−αT , (8) where T = n∑ i=1 log[1− [1− (1− e −λ xi )θ]2]−1. (9) And A1 is the normalizing constant and A−1 1 = ∫ ∞ 0 αn−1e−αTdα, A−1 1 = Γ(n) Tn . Therefore A1 = Tn Γ(n) . Using the value of A1 in Equation (8), we have π1(α|x) = Tn Γ(n) αn−1e−αT . (10) Now, we will estimate (α) with Jeffrey’s prior under three Loss Functions as follow- ing: • The Relative Quadratic (Rq) Loss Function was introduced by [14] is given by r(α̂) = ( α̂−α α )2 (11) • The precautionary (P) Loss Function was introduced by [15] is given by r(α̂) = (α̂− α)2 α̂ (12) Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 5 of 27 • The Entropy (E) Loss Function, see [16] r(α̂) = ( α̂ α − log ( α̂ α ) − 1 ) , (13) (a) Estimation under Relative Quadratic Loss Function Theorem 1. Assuming the Rq loss function, the Bayesian estimator of α, if λ and θ are known, is of the form α̂Rq = n− 2 T (14) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the Rq loss function in Equation (11) is given by r(α̂) = ∫ ∞ 0 ( α̂−α α )2 π1(α|x)dα, (15) By substituting Equation (10) in Equation (15), we get r(α̂) = Tn Γ(n) ∫ ∞ 0 ( α̂−α α )2 αn−1e−αTdα, by setting z = αT r(α̂) = 1 Γ(n) [ α̂2T 2 ∫ ∞ 0 zn−3e−zdz − 2α̂T ∫ ∞ 0 zn−2e−zdz + ∫ ∞ 0 zn−1e−zdz ] after solving the integral, we get r(α̂) = [ α̂2T 2 (n− 1)(n− 2) − 2α̂T n− 1 + 1 ] (16) The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ α̂2T 2 (n− 1)(n− 2) − 2α̂T n− 1 + 1 ] = 0, α̂Rq = n− 2 T . Where T is defined in Equation (9) Hence, the theorem is proved. (b) Estimation under Precautionary Loss Function Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 6 of 27 Theorem 2. Assuming the P loss function, the Bayesian estimator of α, if λ and θ are known, is of the form α̂p = (n(n+ 1)) 1 2 T (17) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the P loss function in Equa- tion (12) is given by r(α̂) = ∫ ∞ 0 (α̂− α)2 α̂ π1(α|x)dα. (18) By substituting Equation (10) in Equation (18), we get r(α̂) = Tn Γ(n) ∫ ∞ 0 (α̂− α)2 α̂ αn−1e−αTdα, r(α̂) = Tn Γ(n) [∫ ∞ 0 α̂αn−1 e−αTdα− 2 ∫ ∞ 0 αne−αTdα + 1 α̂ ∫ ∞ 0 αn+1e−αTdα ] by setting z = αT r(α̂) = 1 Γ(n) [ α̂ ∫ ∞ 0 zn−1e−zdz − 2 1 T ∫ ∞ 0 zne−zdz + 1 α̂ T 2 ∫ ∞ 0 zn+1e−zdz ] after solving the integral, we get r(α̂) = [ α̂ − 2 n T + n(n+ 1) α̂ T 2 ] (19) The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ α̂ − 2 n T + n(n+ 1) α̂ T 2 ] = 0, α̂p = (n(n+ 1)) 1 2 T . Where T is defined in Equation (9) Hence, the theorem is proved. (c) Estimation under Entropy Loss Function Theorem 3. Assuming the E loss function, the Bayesian estimator of α, if Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 7 of 27 λ and θ are known, is of the form α̂E = (n− 1) T (20) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the E loss function in Equa- tion (13) is given by r(α̂) = ∫ ∞ 0 ( α̂ α − log ( α̂ α ) − 1 ) π1(α|x)dα, (21) By substituting Equation (10) in Equation (21), we get r(α̂) = Tn Γ(n) ∫ ∞ 0 [ α̂ α − log( α̂ α )− 1 ] αn−1e−αTdα = Tn Γ(n) [ α̂ ∫ ∞ 0 αn−2 e−αTdα− log(α̂) ∫ ∞ 0 αn−1e−αTdα + ∫ ∞ 0 log(α)αn−1e−αTdα − ∫ ∞ 0 αn−1e−αTdα ] by setting z = αT r(α̂) = 1 Γ(n) [ α̂ T ∫ ∞ 0 zn−2e−zdz − log(α̂) ∫ ∞ 0 zn−1 e−zdz + ∫ ∞ 0 log( z T ) zn−1e−zdz − ∫ ∞ 0 zn−1e−zdz ] = 1 Γ(n) [ α̂ T ∫ ∞ 0 zn−2e−zdz − log(α̂) ∫ ∞ 0 zn−1e−zdz + ∫ ∞ 0 log(z) zn−1e−zdz − ∫ ∞ 0 log(T ) zn−1e−zdz − ∫ ∞ 0 zn−1e−zdz ] , After solving the integral, we get r(α̂) = α̂ T n− 1 − log(α̂) + ψ(n) − log(T ) − 1 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 8 of 27 where ψ(n) = Γ′(n) Γ(n) called Psi (or digamma) function. The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ T n− 1 − log(α̂) + ψ(n) − log(T ) − 1 ] = 0, α̂E = (n− 1) T . Where T is defined in Equation (9) Hence, the theorem is proved. (ii) Posterior Distribution under Quasi prior The Quasi prior for parameter α is given by g2(α) ∝ 1 αd α, d > 0. (22) By using the Bayes theorem, we get π2(α|x) ∝ L(x|α)g2(α). (23) By substituting Equation (4) and Equation (22) in Equation (23), we have π2(α|x) = A2α n−dexp[−α n∑ i=1 log[1− [1− (1− e −λ xi )θ]2]−1]. π2(α|x) = A2α n−de−αT , (24) where T is defined in Equation (9). And A2 is the normalizing constant and A−1 2 = ∫ ∞ 0 αn−de−αTdα, A−1 2 = Γ(n− d+ 1) Tn−d+1 . Therefore A2 = Tn−d+1 Γ(n− d+ 1) . Using the value of A2 in Equations (24), we have π2(α|x) = αn−dTn−d+1 Γ(n− d+ 1) e−αT . (25) Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 9 of 27 (i) Estimation under Relative Quadratic Loss Function Theorem 4. Assuming the Rq loss function, the Bayesian estimator of α, if λ and θ is known, are of the form α̂Rq = n− d− 1 T (26) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the Rq loss function in Equation (11) is given by r(α̂) = ∫ ∞ 0 ( α̂−α α )2 π2(α|x)dα, (27) By substituting Equation (25) in Equation (27), we get r(α̂) = Tn−d+1 Γ(n− d+ 1) ∫ ∞ 0 ( α̂−α α )2 αn−de−αTdα = Tn−d+1 Γ(n− d+ 1) ∫ ∞ 0 [ α̂2αn−d−2 − 2 α̂αn−d−1 + αn−d ] e−αTdα By setting z = αT r(α̂) = Tn−d Γ(n− d+ 1) ∫ ∞ 0 [ α̂2 z n−d−2 Tn−d−2 − 2α̂ zn−d−1 Tn−d−1 + zn−d Tn−d ] e−zdz After solving the integral, we get r(α̂) = [ α̂2T 2 (n− d)(n− d− 1) − 2α̂T n− d + 1 ] (28) The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ α̂2T 2 (n− d)(n− d− 1) − 2α̂T n− d + 1 ] = 0, α̂Rq = n− d− 1 T . Where T is defined in Equation (9). Hence, the theorem is proved. (ii) Estimation under Precautionary Loss Function Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 10 of 27 Theorem 5. Assuming the P loss function, the Bayesian estimator of α, if λ and θ are known, is of the form α̂P = ((n− d+ 2)(n− d+ 1)) 1 2 T (29) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the P loss function in Equation (12) is given by r(α̂) = ∫ ∞ 0 (α̂− α)2 α̂ π2(α|x)dα, (30) By substituting Equation (25) in Equation (30), we get r(α̂) = Tn−d+1 Γ(n− d+ 1) ∫ ∞ 0 (α̂− α)2 α̂ αn−de−αTdα = Tn−d+1 Γ(n− d+ 1) [ α̂ ∫ ∞ 0 αn−d e−αTdα− 2 ∫ ∞ 0 αn−d+1e−αTdα + 1 α̂ ∫ ∞ 0 αn−d+2e−αTdα ] By setting z = αT r(α̂) = 1 Γ(n− d+ 1) [ α̂ ∫ ∞ 0 zn−de−zdz − 2 1 T ∫ ∞ 0 zn−d+1e−zdz + 1 α̂ T 2 ∫ ∞ 0 zn−d+2e−zdz ] After solving the integral, we get r(α̂) = [ α̂ − 2 n− d+ 1 T + (n− d+ 2)(n− d+ 1) α̂ T 2 ] The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ α̂ − 2 n− d+ 1 T + (n− d+ 2)(n− d+ 1) α̂ T 2 ] = 0, Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 11 of 27 α̂P = ((n− d+ 2)(n− d+ 1)) 1 2 T . Where T is defined in Equation (9). Hence, the theorem is proved. (iii) Estimation under Entropy Loss Function Theorem 6. Assuming the E loss function, the Bayesian estimator of α, if λ and θ are known, is of the form α̂E = (n− d) T (31) Where T = ∑n i=1 log[1− [1− (1− e −λ xi )θ]2]−1. Proof. The risk function of the estimator α under the E loss function in Equation (13) is given by r(α̂) = ∫ ∞ 0 ( α̂ α − log ( α̂ α ) − 1 ) π2(α|x)dα, (32) By substituting Equation (25) in Equation (32). r(α̂) = Tn−d+1 Γ(n− d+ 1) ∫ ∞ 0 [ α̂ α − log( α̂ α )− 1 ] αn−de−αTdα = Tn−d+1 Γ(n− d+ 1) [ α̂ ∫ ∞ 0 αn−d−1 e−αTdα − log(α̂) ∫ ∞ 0 αn−de−αTdα + ∫ ∞ 0 log(α)αn−de−αTdα − ∫ ∞ 0 αn−de−αTdα ] By setting z = αT r∗(α̂) = 1 Γ(n− d+ 1) [ α̂ T ∫ ∞ 0 zn−d−1e−zdz − log(α̂) ∫ ∞ 0 zn−de−zdz + ∫ ∞ 0 log( z T ) zn−de−zdz − ∫ ∞ 0 zn−de−zdz ] , = 1 Γ(n− d+ 1) [ α̂ T ∫ ∞ 0 zn−d−1e−zdz − log(α̂) ∫ ∞ 0 zn−de−zdz + ∫ ∞ 0 log(z) zn−de−zdz − ∫ ∞ 0 log(T ) zn−de−zdz − ∫ ∞ 0 zn−de−zdz ] , Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 12 of 27 After solving the integral, we get r(α̂) = α̂ T n− d − log(α̂) + ψ(n− d+ 1) − log(T ) − 1, where ψ(n− d+ 1) = Γ′(n−d+1) Γ(n−d+1) called Psi (or digamma) function. The optimal estimator is obtained by solving ∂r(α̂,T ) ∂α̂ = 0 ∂ ∂α̂ [ T n− d − log(α̂) + ψ(n− d+ 1) − log(T ) − 1 ] = 0, α̂E = (n− d) T . Where T is defined in Equation (9). Hence, the theorem is proved. 3.2. Case 2: standard Bayes Estimation Suppose α is unknown and λ, θ are known. And α has Gamma prior distribution α ∼ Gamma(a,b); thus, the prior for α is given by: π(α) = ba Γ(a) αa−1e−bα, α > 0, a, b > 0. (33) By combining (4) and (33) the posterior distribution of the unknown parameter α is given by: π∗(α|x) = k αn+a−1e−α[b− ∑n i=1 log[1−[1−[1−e −λ xi ]θ]2]], (34) where k is the normalizing constant, can be written as k−1 = ∫ ∞ 0 αn+a−1e−α[b− ∑n i=1 log[1−[1−[1−e −λ xi ]θ]2]] dα. (35) Now, we will estimate (α) with Gamma prior under three Loss Functions as following: (i) Squared Error loss function (SE) when α is unknown The SE loss function of ϕ(α) can be defined as: ϕ̂SE(α) = E(ϕ(α)|x) = ∫ ϕ(α)π∗(α|x)dα, (36) then the Bayes estimator of ϕ(α) under SE loss function, denoted by ϕ̂SE(α), and Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 13 of 27 can be found using Equations (34) and (36). ϕ̂SE(α) = k ∫ ∞ 0 ϕ(α)αn+a−1e−α[b− ∑n i=1 log[1−[1−[1−e −λ xi ]θ]2]] dα. (37) Where k−1is defined in equation (35). (ii) LINEX loss function (LINEX) when α is unknown The LINEX loss function proposed by [17] as follows: ϕ̂LINEX(α) = −1 τ ln [ Eα(e −τϕ(α)|x) ] = −1 τ ln [∫ e−τϕ(α)π∗(α|x)dα ] , (38) then the Bayes estimator of ϕ(α) under the LINEX loss function, denoted by ϕ̂LINEX(α), can be found by using Equations (34) and (38). ϕ̂LINEX(α) = −1 τ ln [ k ∫ ∞ 0 e−τϕ(α)αn+a−1e−α[b− ∑n i=1 log[1−[1−[1−e −λ xi ]θ]2]] dα ] , (39) Where k−1is defined in equation (35). (iii) General Entropy loss function (GE) when α is unknown The GE loss function for ϕ(α) can be defined by [16] as follows: ϕ̂GE(α) = [ Eα(ϕ(α) −q|x) ]−1 q = [∫ ϕ(α)−qπ∗(α|x)dα ]−1 q , (40) then the Bayes estimator of ϕ(α) under the GE loss function, denoted by ϕ̂GE(α), can be found by using Equations (34) and (40). ϕ̂GE(α) = [ k ∫ ∞ 0 ϕ(α)−qαn+a−1e−α[b− ∑n i=1 log[1−[1−[1−e −λ xi ]θ]2]] dα ]−1 q , (41) Where k−1is defined in equation (35). The Bayes estimators of α under SE, LINEX, GE loss functions are found numeri- cally by the NIntegrate function via Mathematica 11.3 4. Simulation Study of Bayes and ML Estimators In this section, numerical results of Bayes and ML estimators for parameter (α) of TIITL-GIED have been presented using Monte Carlo simulation. Comparing between Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 14 of 27 these estimates are studied based on FindRoot and NIntegrate functions in Mathematica (V.11.3) program. Also, the performance of the parameters is determined according to the bias and mean squared error (MSE), that given, respectively, by Bias(β̂) = E(β̂)− β, (42) and MSE(β̂) = E(β̂ − β)2. (43) The following steps are used to accomplish this process. (i) A sample of size n from TIITL-GIED is generated using Equation (3) for given values of the parameters. (ii) Based on ML and Bayes methods and for different initial values of parameters, the ML and Bayesian estimates of parameter α under Rq, P and E loss functions is computed by solving the Equations (14), (17), (20), (26), (29) and (31). (iii) Based on ML and standard Bayes and for given values of hyperparameters (a = 1, b = 1) and (a = 1, b = 2), the ML and Bayesian estimates of parameter α is computed under SE, LINEX and GE loss functions by solving the Equations (37), (39) and (41). (iv) Based on ML and Bayes methods and for different initial values of parameters, the ML and Bayesian estimates of parameter α under all loss functions that we include in this study is computed by solving the Equations (14), (17), (20), (26), (29), (31), (37), (39) and (41). (v) The proposed values of all MLE and Bayes of the shape parameters are used. (vi) The performance of the parameters is determined according to the Bias and MSE using Equations (42) and (43). (vii) The above steps are repeated 1000 times. Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 15 of 27 Table 1: The ML and Bayesian estimates of the unknown parameter α under Jeffery and Quasi priors for the initial values (α = 1,θ = 1, λ = 1). n Parameters ML Estimates (Bias) (MSE) Jeffery’s prior Quasi prior α̂Rq α̂P α̂E α̂Rq α̂P α̂E D=0.3 D=1 D=0.3 D=1 D=0.3 D=1 10 α 1.09649 0.87719 1.15000 0.98684 0.95394 0.87719 1.22684 1.15000 1.06359 0.98684 0.09649 -0.12281 0.15000 -0.01316 -0.04606 -0.12281 0.022684 0.15000 0.06359 -0.01316 0.14407 0.10133 0.17073 0.10933 0.10412 0.10133 0.22016 0.17073 0.13084 0.10933 30 α 1.03119 0.96244 1.04823 0.99681 0.98650 0.96244 1.07229 1.04823 1.02087 0.99681 0.03119 -0.03756 0.04823 -0.00319 -0.01350 -0.03756 0.07230 0.04823 0.02087 -0.00319 0.03826 0.03390 0.04086 0.03486 0.03431 0.03390 0.04555 0.04086 0.03699 0.03486 50 α 1.02204 0.98116 1.03221 1.00160 0.99547 0.98116 1.04652 1.03221 1.01591 1.00160 0.02204 -0.01884 0.03221 0.00160 -0.00453 -0.01884 0.04652 0.03221 0.01591 0.00160 0.02232 0.02048 0.02331 0.02098 0.02074 0.02048 0.02506 0.02331 0.02182 0.02098 70 α 1.01599 0.98696 1.02322 1.00147 0.99712 0.98696 1.03338 1.02322 1.01163 1.00147 0.01599 -0.01304 0.02322 0.00147 -0.00288 -0.01304 0.03338 0.02322 0.01163 0.00147 0.01559 0.01464 0.01609 0.01490 0.01477 0.01464 0.01697 0.01609 0.01533 0.01490 Table 2: The ML and Bayesian estimates of the unknown parameter α under Jeffery and Quasi priors for the initial values (α = 2,θ = 2, λ = 2). n Parameters ML Estimates (Bias) (MSE) Jeffery’s prior Quasi prior α̂Rq α̂P α̂E α̂Rq α̂P α̂E D=0.3 D=1 D=0.3 D=1 D=0.3 D=1 10 α 2.21363 1.77091 2.32168 1.99227 1.92586 1.77091 2.47680 2.32168 2.14722 1.99227 0.21363 -0.22909 0.32168 -0.00773 -0.07414 -0.22909 0.47680 0.32168 0.14722 -0.00773 0.61962 0.41983 0.73486 0.46498 0.43994 0.41983 0.94590 0.73486 0.56173 0.46498 30 α 2.06463 1.92698 2.09875 1.99580 1.97516 1.92698 2.14693 2.09875 2.04398 1.99580 0.06463 -0.07302 0.09875 0.00420 -0.02484 -0.07302 0.14693 0.09875 0.04398 0.00420 0.14499 0.12799 0.15525 0.13160 0.12949 0.12799 0.17385 0.15525 0.13994 0.13160 50 α 2.03165 1.95038 2.05186 1.99101 1.97882 1.95038 2.08031 2.05186 2.01946 1.99101 0.03165 -0.04962 0.05186 -0.00899 -0.02118 -0.04962 0.08031 0.05186 0.01946 -0.00899 0.08375 0.07872 0.08709 0.07955 0.07895 0.07872 0.09321 0.08709 0.08214 0.07955 70 α 2.03025 1.97224 2.04470 2.00124 1.99254 1.97224 2.06500 2.04470 2.02155 2.00124 0.03025 -0.02776 0.04470 0.00124 -0.00746 -0.02776 0.06500 0.04470 0.02155 0.00124 0.06242 0.05881 0.06438 0.05976 0.05930 0.05881 0.06786 0.06438 0.06145 0.05976 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 16 of 27 Table 3: The ML and Bayesian estimates of the unknown parameter α via the standard Bayes techniques for the initial values (α = 1, θ = 1, λ = 1) and (a=1,b=1). n Parameters ML Estimates (Bias) (MSE) Bayes Estimates SE (Bias) (MSE) LINEX (Bias) (MSE) GE (Bias) (MSE) τ=0.001 τ=3 τ=5 q = -1 q = 3 q = -3 10 α 1.10296 1.08190 1.08190 0.98133 0.86996 1.08190 0.88153 1.17752 0.10296 0.08190 0.08190 -0.01867 -0.13004 0.08190 -0.11847 0.17752 0.15134 0.10980 0.10980 0.06902 0.05968 0.10980 0.08247 0.15362 30 α 1.03901 1.03660 1.03660 1.00245 0.95656 1.03660 0.96932 1.06967 0.03901 0.03660 0.03660 0.00245 -0.04344 0.03660 -0.03068 0.06967 0.03769 0.03477 0.03477 0.02918 0.02608 0.03477 0.03018 0.04046 50 α 1.02008 1.01926 1.01926 0.99907 0.97070 1.01926 0.97917 1.03913 0.02008 0.01928 0.01928 -0.00093 -0.02930 0.01928 -0.02083 0.03913 0.02128 0.02036 0.02036 0.01844 0.01730 0.02036 0.01888 0.02230 70 α 1.01138 1.01101 1.01101 0.99669 0.97620 1.01101 0.98247 1.02519 0.01138 0.01101 0.01101 -0.00331 -0.02380 0.01101 -0.01755 0.02519 0.01516 0.01471 0.01471 0.01378 0.01325 0.01471 0.01408 0.01563 Table 4: The ML and Bayesian estimates of the unknown parameter α via the standard Bayes techniques for the initial values (α = 0.7, θ = 0.5, λ = 0.8) and (a=1,b=2). n Parameters ML Estimates (Bias) (MSE) Bayes Estimates SE (Bias) (MSE) LINEX (Bias) (MSE) GE (Bias) (MSE) τ=0.001 τ=3 τ=5 q = -1 q = 3 q = -3 10 α 0.76742 0.72191 0.72191 0.67520 0.61879 0.72191 0.58822 0.78572 0.06742 0.02191 0.02191 -0.02480 -0.08121 0.02191 -0.11178 0.08572 0.07785 0.04536 0.04536 0.03452 0.03044 0.04536 0.04229 0.06052 30 α 0.72302 0.71162 0.71162 0.69528 0.67259 0.71162 0.66545 0.73434 0.02302 0.01162 0.01162 -0.00472 -0.02741 0.01162 -0.03455 0.03434 0.01982 0.01690 0.01690 0.01524 0.01402 0.01690 0.01585 0.01903 50 α 0.71283 0.70656 0.70656 0.69679 0.68279 0.70656 0.67876 0.72033 0.01283 0.00656 0.00656 -0.00321 -0.01722 0.00656 -0.02124 0.02033 0.00984 0.00900 0.00900 0.00847 0.00809 0.00900 0.00872 0.00972 70 α 0.70903 0.70469 0.70469 0.69770 0.68755 0.70469 0.68479 0.71457 0.00903 0.00469 0.00469 -0.00230 -0.01245 0.00469 -0.01521 0.01457 0.00705 0.00662 0.00662 0.00635 0.00614 0.00662 0.00647 0.00691 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 17 of 27 Ta bl e 5: T he M L an d B ay es ia n es tim at es of th e un kn ow n pa ra m et er α fo r th e in iti al va lu es (α = 1 ,θ = 1 ,λ = 1 ) an d (a = 1, b= 1) . n Pa ra m et er s M L Es tim at es (B ia s) (M SE ) Je ffe ry ’s pr io r Q ua si pr io r SE (B ia s) (M SE ) LI N EX (B ia s) (M SE ) G E (B ia s) (M SE ) α̂ R q α̂ P α̂ E α̂ R q α̂ P α̂ E τ = 0. 00 1 τ = 3 τ = 5 q= -1 q= 3 q= -3 D = 0. 3 D = 1 D = 0. 3 D = 1 D = 0. 3 D = 1 10 α 1. 12 61 3 0. 90 09 0 1. 18 10 9 1. 01 35 2 0. 97 97 3 0. 90 09 0 1. 26 00 1 1. 18 10 9 1. 09 23 4 1. 01 35 2 1. 09 95 0 1. 09 95 0 0. 99 46 4 0. 87 97 1 1. 09 95 0 0. 89 59 2 1. 19 67 3 0. 12 61 3 -0 .0 99 10 0. 18 10 9 0. 01 35 2 -0 .0 20 27 -0 .0 99 10 0. 26 00 1 0. 18 10 9 0. 09 23 4 0. 01 35 2 0. 09 95 0 0. 09 95 0 -0 .0 05 36 -0 .1 20 29 0. 09 95 0 -0 .1 04 08 0. 19 67 3 0. 20 07 6 0. 12 81 3 0. 23 61 3 0. 14 99 1 0. 14 03 3 0. 12 81 3 0. 29 90 2 0. 23 61 3 0. 18 24 5 0. 14 99 1 0. 13 84 0 0. 13 84 0 0. 08 35 6 0. 06 55 7 0. 13 84 0 0. 09 61 7 0. 19 09 6 30 α 1. 02 90 6 0. 96 04 6 1. 04 60 8 0. 99 47 6 0. 98 44 7 0. 96 04 6 1. 07 00 9 1. 04 60 8 1. 01 87 7 0. 99 47 6 1. 02 69 0 1. 02 69 0 0. 99 34 0 0. 94 83 0 1. 02 69 0 0. 96 03 3 1. 05 97 5 0. 02 90 7 -0 .0 39 54 0. 04 60 8 -0 .0 05 24 -0 .0 01 55 -0 .0 39 54 0. 07 00 9 0. 04 60 8 0. 01 87 7 -0 .0 05 24 0. 02 69 0 0. 02 69 0 -0 .0 06 58 -0 .0 51 70 0. 02 69 0 -0 .0 39 70 0. 05 97 5 0. 03 74 3 0. 03 34 3 0. 03 99 3 0. 03 42 1 0. 03 37 2 0. 03 34 3 0. 04 44 7 0. 03 99 3 0. 03 62 1 0. 03 42 1 0. 03 45 0 0. 03 45 0 0. 02 95 6 0. 02 71 0 0. 03 45 0 0. 03 11 0 0. 03 95 8 50 α 1. 01 32 3 0. 97 27 0 1. 02 33 2 0. 99 29 7 0. 98 68 9 0. 97 27 0 1. 03 75 0 1. 02 33 2 1. 00 71 5 0. 99 29 7 1. 01 26 0 1. 01 26 0 0. 99 26 3 0. 96 46 1 1. 01 26 0 0. 97 27 3 1. 03 23 0 0. 01 32 3 -0 .0 27 30 0. 02 33 2 -0 .0 07 03 -0 .0 13 11 -0 .0 27 30 0. 03 75 0 0. 02 33 2 0. 00 71 5 -0 .0 07 03 0. 01 26 0 0. 01 26 0 -0 .0 07 37 -0 .0 35 39 0. 01 26 0 -0 .0 27 27 0. 03 23 0 0. 02 09 0 0. 01 98 4 0. 02 16 8 0. 01 99 5 0. 01 98 3 0. 01 98 4 0. 02 31 3 0. 02 16 8 0. 02 05 2 0. 01 99 5 0. 01 99 9 0. 01 99 9 0. 01 83 53 0. 01 75 7 0. 01 99 9 0. 01 90 5 0. 02 16 6 70 α 1. 01 66 4 0. 98 76 0 1. 02 38 8 1. 00 21 2 0. 99 77 6 0. 98 76 0 1. 03 40 5 1. 02 38 8 1. 01 22 9 1. 00 21 2 1. 01 62 0 1. 01 62 0 1. 00 17 5 0. 98 10 7 1. 01 62 0 0. 98 75 1 1. 03 04 5 0. 01 66 5 -0 .0 12 40 0. 02 38 8 0. 00 21 2 -0 .0 02 24 -0 .0 12 40 0. 03 40 5 0. 02 38 8 0. 01 22 9 0. 00 21 2 0. 01 62 0 0. 01 62 0 0. 00 17 5 -0 .0 18 93 0. 01 62 0 -0 .0 12 49 0. 03 04 5 0. 01 45 7 0. 01 36 4 0. 01 50 7 0. 01 38 9 0. 01 37 7 0. 01 36 4 0. 01 59 4 0. 01 50 7 0. 01 43 2 0. 01 38 9 0. 01 41 2 0. 01 41 2 0. 01 30 8 0. 01 23 9 0. 01 41 2 0. 01 32 4 0. 01 51 8 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 18 of 27 Ta bl e 6: T he M L an d B ay es ia n es tim at es of th e un kn ow n pa ra m et er α fo r th e in iti al va lu es (α = 0 .7 ,θ = 0 .5 ,λ = 0 .8 ) an d (a = 3, b= 5) . n Pa ra m et er s M L Es tim at es (B ia s) (M SE ) Je ffe ry ’s pr io r Q ua si pr io r SE (B ia s) (M SE ) LI N EX (B ia s) (M SE ) G E (B ia s) (M SE ) α̂ R q α̂ P α̂ E α̂ R q α̂ P α̂ E τ = 0. 00 1 τ = 3 τ = 5 q= -1 q= 3 q= -3 D = 0. 3 D = 1 D = 0. 3 D = 1 D = 0. 3 D = 1 10 α 0. 78 76 7 0. 63 01 3 0. 82 61 1 0. 70 89 1 0. 68 52 7 0. 63 01 3 0. 88 13 1 0. 82 61 1 0. 76 40 4 0. 70 89 1 0. 71 76 8 0. 71 76 8 0. 67 89 6 0. 63 03 9 0. 71 76 8 0. 60 56 1 0. 77 15 9 0. 08 76 7 -0 .0 69 87 0. 12 61 1 0. 00 89 0 -0 .0 14 73 -0 .0 69 87 0. 18 13 1 0. 12 61 1 0. 06 40 4 0. 00 89 0 0. 01 76 8 0. 01 76 8 -0 .0 21 04 -0 .0 69 61 0. 01 76 8 -0 .0 94 39 0. 07 15 9 0. 08 54 2 0. 05 46 3 0. 10 14 1 0. 06 30 4 0. 05 90 5 0. 05 46 3 0. 13 01 8 0. 10 14 1 0. 07 72 4 0. 06 30 4 0. 02 95 9 0. 02 95 9 0. 02 38 4 0. 02 23 1 0. 02 95 9 0. 02 97 6 0. 03 89 7 30 α 0. 72 28 2 0. 67 46 3 0. 73 47 6 0. 69 87 2 0. 69 14 9 0. 67 46 3 0. 75 16 3 0. 73 47 6 0. 71 55 9 0. 69 87 2 0. 70 72 1 0. 70 72 1 0. 69 20 9 0. 67 09 7 0. 70 72 1 0. 66 41 2 0. 72 84 4 0. 02 28 2 -0 .0 25 37 0. 03 47 6 -0 .0 01 28 -0 .0 08 51 -0 .0 25 37 0. 05 16 3 0. 03 47 6 0. 01 55 9 -0 .0 01 28 0. 00 72 1 0. 00 72 1 -0 .0 07 91 -0 .0 29 03 0. 00 72 1 -0 .0 35 88 0. 02 84 3 0. 01 91 8 0. 01 68 9 0. 02 04 9 0. 01 74 3 0. 01 71 5 0. 01 68 9 0. 02 28 4 0. 02 04 9 0. 01 85 3 0. 01 74 3 0. 01 39 8 0. 01 39 8 0. 01 28 2 0. 01 21 0 0. 01 39 8 0. 01 35 7 0. 01 55 9 50 α 0. 71 53 7 0. 68 67 6 0. 72 24 9 0. 70 10 6 0. 69 67 7 0. 68 67 6 0. 73 25 1 0. 72 24 9 0. 71 10 8 0. 70 10 6 0. 70 67 7 0. 70 67 7 0. 69 73 5 0. 68 38 4 0. 70 67 7 0. 68 00 1 0. 72 00 2 0. 01 53 7 -0 .0 13 24 0. 02 24 9 0. 00 10 6 -0 .0 03 23 -0 .0 13 24 0. 03 25 1 0. 02 24 9 0. 01 10 8 0. 00 10 6 0. 00 67 7 0. 00 67 7 -0 .0 02 65 -0 .0 16 16 0. 00 67 7 -0 .0 19 99 0. 02 00 2 0. 01 07 0 0. 00 98 2 0. 01 11 8 0. 01 00 5 0. 00 99 4 0. 00 98 2 0. 01 20 3 0. 01 11 8 0. 01 04 6 0. 01 00 5 0. 00 89 0 0. 00 89 0 0. 00 84 0 0. 00 80 2 0. 00 89 0 0. 00 85 9 0. 00 95 9 70 α 0. 71 02 1 0. 68 99 2 0. 71 52 7 0. 70 00 7 0. 69 70 2 0. 68 99 2 0. 72 23 7 0. 71 52 7 0. 70 71 7 0. 70 00 7 0. 70 44 5 0. 70 44 5 0. 69 76 6 0. 68 77 9 0. 70 44 5 0. 68 51 0 0. 71 40 6 0. 01 02 1 -0 .0 10 08 0. 01 52 7 0. 00 00 6 -0 .0 02 98 -0 .0 10 08 0. 02 23 7 0. 01 52 7 0. 00 71 7 0. 00 00 6 0. 00 44 5 0. 00 44 5 -0 .0 02 34 -0 .0 12 21 0. 00 44 5 -0 .0 14 90 0. 01 40 6 0. 00 69 8 0. 00 65 9 0. 00 72 0 0. 00 66 8 0. 00 66 3 0. 00 65 9 0. 00 76 1 0. 00 72 0 0. 00 68 7 0. 00 66 8 0. 00 61 3 0. 00 61 3 0. 00 58 9 0. 00 57 1 0. 00 61 3 0. 00 60 0 0. 00 64 8 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 19 of 27 Ta bl e 7: T he M L an d B ay es ia n es tim at es of th e un kn ow n pa ra m et er α fo r th e in iti al va lu es (α = 0 .7 ,θ = 0 .5 ,λ = 0 .8 ) an d (a = 1, b= 1) . n Pa ra m et er s M L Es tim at es (B ia s) (M SE ) Je ffe ry ’s pr io r Q ua si pr io r SE (B ia s) (M SE ) LI N EX (B ia s) (M SE ) G E (B ia s) (M SE ) α̂ R q α̂ P α̂ E α̂ R q α̂ P α̂ E τ = 0. 00 1 τ = 3 τ = 5 q= -1 q= 3 q= -3 D = 0. 3 D = 1 D = 0. 3 D = 1 D = 0. 3 D = 1 10 α 0. 76 93 2 0. 61 54 5 0. 80 68 7 0. 69 23 9 0. 66 93 1 0. 61 54 5 0. 86 07 8 0. 80 68 7 0. 74 62 4 0. 69 23 9 0. 77 97 7 0. 77 97 7 0. 72 51 6 0. 66 04 0 0. 77 97 7 0. 63 53 6 0. 84 86 9 0. 06 93 2 -0 .0 84 55 0. 10 68 7 -0 .0 07 62 -0 .0 30 69 -0 .0 84 55 0. 16 07 8 0. 10 68 7 0. 04 62 4 -0 .0 07 62 0. 07 97 7 0. 07 97 7 0. 02 51 6 -0 .0 39 60 0. 07 97 7 -0 .0 64 64 0. 14 86 9 0. 07 51 8 0. 05 21 9 0. 08 88 3 0. 05 70 6 0. 05 42 1 0. 05 21 9 0. 11 39 5 0. 08 88 3 0. 06 83 5 0. 05 70 6 0. 06 61 8 0. 06 61 8 0. 04 48 0 0. 03 19 6 0. 06 61 8 0. 04 38 9 0. 09 29 7 30 α 0. 72 26 3 0. 67 44 5 0. 73 45 7 0. 69 85 4 0. 69 13 1 0. 67 44 5 0. 75 14 4 0. 73 45 7 0. 71 54 0 0. 69 85 4 0. 72 85 6 0. 72 85 6 0. 71 14 3 0. 68 77 0 0. 72 85 6 0. 68 12 9 0. 75 18 2 0. 02 26 3 -0 .0 25 55 0. 03 45 7 -0 .0 01 46 -0 .0 08 69 -0 .0 25 55 0. 05 14 4 0. 03 45 7 0. 01 54 0 -0 .0 01 46 0. 02 85 5 0. 02 85 5 0. 01 14 3 -0 .0 12 30 0. 02 85 5 -0 .0 18 71 0. 05 18 2 0. 01 90 2 0. 01 67 8 0. 02 03 2 0. 01 73 0 0. 01 70 2 0. 01 67 8 0. 02 26 6 0. 02 03 2 0. 01 83 8 0. 01 73 0 0. 01 86 6 0. 01 86 6 0. 01 63 0 0. 01 42 3 0. 01 86 6 0. 01 59 5 0. 02 16 8 50 α 0. 71 51 3 0. 68 65 3 0. 72 22 5 0. 70 08 3 0. 69 65 4 0. 68 65 3 0. 73 22 6 0. 72 22 5 0. 71 08 4 0. 70 08 3 0. 71 89 3 0. 71 89 3 0. 70 87 9 0. 69 42 8 0. 71 89 3 0. 69 06 5 0. 73 29 5 0. 01 51 3 -0 .0 13 47 0. 02 22 5 0. 00 08 3 -0 .0 03 46 -0 .0 13 47 0. 03 22 6 0. 02 22 5 01 08 42 8 0. 00 08 3 0. 01 89 3 0. 01 89 3 0. 00 87 9 -0 .0 05 72 0. 01 89 3 -0 .0 09 35 0. 03 29 5 0. 01 10 7 0. 01 01 7 0. 01 15 5 0. 01 04 1 0. 01 03 0 0. 01 01 7 0. 01 24 1 0. 01 15 5 0. 01 08 3 0. 01 04 1 0. 01 09 9 0. 01 09 9 0. 01 01 1 0. 00 92 6 0. 01 09 9 0. 00 99 0 0. 01 21 4 70 α 0. 70 65 5 0. 68 63 6 0. 71 15 8 0. 69 64 6 0. 69 34 3 0. 68 63 6 0. 71 86 5 0. 71 15 8 0. 70 35 2 0. 69 64 6 0. 70 93 8 0. 70 93 8 0. 70 22 9 0. 69 19 9 0. 70 93 8 0. 68 93 5 0. 71 93 3 0. 00 65 5 -0 .0 13 64 0. 01 15 8 -0 .0 03 54 -0 .0 06 57 -0 .0 13 64 0. 01 86 5 0. 01 15 8 0. 00 35 2 -0 .0 03 54 0. 00 93 8 0. 00 93 8 0. 00 22 9 -0 .0 08 00 0. 00 93 8 -0 .0 10 65 0. 01 93 3 0. 00 73 9 0. 00 71 2 0. 00 75 8 0. 00 71 5 0. 00 71 2 0. 00 71 2 0. 00 79 5 0. 00 75 8 0. 00 72 9 0. 00 71 5 0. 00 73 4 0. 00 73 4 0. 00 69 7 0. 00 66 3 0. 00 73 4 0. 00 69 6 0. 00 78 3 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 20 of 27 4.1. Results and Remarks The main remarks and findings of the two estimation methods from the above tables summed up as follows. 4.1.1. Results of Bayes and ML Estimates of α from Tables (1, 2). (i) The estimates of α get closer to the actual values with increasing sample size. Fur- ther, the MSEs and bias decrease and sometimes tend to zero. (ii) The MSEs of ML and Bayes estimates of α decrease as sample size increase. (iii) The Bayes estimate under Rq and E loss functions usually gives better estimation than ML estimate when n = 10, 30, 50 and 70. It has the smallest MSEs. (iv) The α̂, bias and MSEs values for Bayes estimates of Rq loss function under Jeffery’s prior and Rq loss function with (d = 1) under Quasi prior are very similar. (v) The α̂, bias and MSEs values for Bayes estimates of P loss function under Jeffery’s prior and P loss function with (d = 1) under Quasi prior are very similar. (vi) The α̂, bias and MSEs values for Bayes estimates of E loss function under Jeffery’s prior and E loss function with (d = 1) under Quasi prior are very similar. (vii) The ML estimate gives better results of MSEs than P loss function under Jeffery’s prior and Quasi prior with (d = 0.3 , 1) (viii) In general Rq loss function under Jeffery’s prior and Rq loss function with (d = 1) under Quasi prior gives better result of MSEs as increase sample size. 4.1.2. Results of standard Bayes and ML Estimates of α from Tables (3, 4). (i) The MSEs of ML and Bayes estimates of α decrease as sample size increase. (ii) The Bayes estimate usually gives better estimation than ML estimate when n = 10, 30, 50 and 70. (iii) The α̂, bias and MSEs values for Bayes estimates under SE loss function, LINEX loss function with (τ = 0.001), and GE loss function with (q = -1), are very similar. (iv) From table 3, the ML estimate gives better results of MSEs than GE loss function with (q = -3). (v) LINEX loss function gives better results of the MSEs as increase the value of τ . Also, GE loss function performs better as we increase the value of q. (vi) In general LINEX loss function with (τ = 5) gives better results of MSE as increase sample size. Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 21 of 27 4.1.3. Results of Bayes and ML Estimates of α from Tables (5 to 7). (i) The MSEs of ML and Bayes estimates of α decrease as sample size increase. (ii) The α̂, bias and MSEs values for Bayes estimates of Rq loss function under Jeffery’s prior and Rq loss function with (d = 1) under Quasi prior are very similar. (iii) The α̂, bias and MSEs values for Bayes estimates of P loss function under Jeffery’s prior and P loss function with (d = 1) under Quasi prior are very similar. (iv) The α̂, bias and MSEs values for Bayes estimates of E loss function under Jeffery’s prior and E loss function with (d = 1) under Quasi prior are very similar. (v) The α̂, bias and MSEs values for Bayes estimates under SE loss function, LINEX loss function with (τ = 0.001), and GE loss function with (q = -1), are very similar. (vi) LINEX loss function gives better results of the MSE as increase the value of τ . Also, GE loss function performs better as increase the value of q. (vii) In general LINEX loss function with (τ = 5) gives better results of MSE as we increase sample size. 5. Applications to Real Data In this section, we applied three real data sets in different fields to demon- strate the utility and flexibility of using the TIITLGIED. The performance of the estimates is evaluated using some information criteria. The best technique is which has the low- est Akaike information criterion (AIC), log-likelihood (LL), Bayesian information criteria (BIC), consistent Akaike information criteria (CAIC), and Hannan-Quinn information cri- terion (HQIC) [18]. The formulas of these criteria are The three data sets are modeled with different distributions in some previous studies. These distributions are the Exponentiated Generalized Inverted Exponential Distribution (EGIED), Generalized Inverted Exponen- tial Distribution (GIED) [19], Exponential distribution (EXPD), Topp Leone Generalized Standard Inverted Exponential Distribution (TIITLGSIED) [4] and Topp Leone Standard Inverted Exponential Distribution (TIITLGSIED) [4]. 5.1. Data set 1 This data is used by [20]. It consists of nonlinear growth curves regarding cold tolerance for 3 plants each from 2 locations (Quebec and Mississippi) and 2 treatments (controlled and chilled) at 7 levels of CO2 concentration. The variable CO2 uptake rate is selected for model fitting. The dataset of 18 observations is recorded as: 10.79, 10.80, 10.86, 10.93, 10.99, 10.96, 10.98, 11.03, 11.08, 11.10, 11.19, 11.25, 11.40, 11.61, 11.69, 11.91, 12.07, and 12.32. Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 22 of 27 Table 8: MLEs of the model parameters and the statistics of the L.L, AIC, BIC, CAIC and HQIC for the data set 1 The model The parameters estimate L.L AIC BIC CAIC HQIC α θ λ TIITLGIED 1371.31 565.29 114.187 -13.785 33.569 36.241 35.284 33.938 GIED —— 53399.4 124.793 -20.185 44.3693 46.1501 45.1693 44.6149 EGIED 1.62187 3.6238 18.929 -49.593 105.187 107.858 106.901 105.555 EXPD 0.08869 —– —– -61.608 125.215 126.105 125.465 125.338 TIITLGSIED 8.88564 0.15370 —– -68.784 141.568 142.368 143.349 141.814 TIITLSIED 0.550591 —– —– -73.958 149.916 150.166 150.806 150.039 Figure 1: The empirical distribution and estimated CDF of models using data set 1. Table 9: Descriptive statistics for data set 1 n Minimum Maximum Mean Median Standard deviation Kurtosis Skewness 18 10.79 12.32 11.2656 11.09 0.4591 2.7667 0.9774 (a) Estimated PDFs. (b) Boxplot of the data set 1. Figure 2: Plots of the estimated PDF of the models and the Boxplot using data set 1. Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 23 of 27 5.2. Data set 2 This data represent current health expenditure in terms of economic growth rates in Saudi Arabia. See [21]. 42.1159410, 44.6173573, 42.4937630, 39.7909737, 35.8400607, 34.1867280 36.1922503, 35.6228733, 29.7100425, 42.9041958, 36.4785600, 37.1177721, 40.1962376, 44.6568298, 52.2795486, 59.9834490, 58.3562946, 62.6256323 57.4845695, 56.8828773. Table 10: MLEs of the model parameters and the statistics of the L.L, AIC, BIC, CAIC and HQIC for the data set 2 The model The parameters estimate L.L AIC BIC CAIC HQIC α θ λ TIITLGIED 4.95268 8.92284 132.01 -73.057 152.114 155.102 153.614 152.698 EGIED 1.19226 38.5462 176.175 -74.122 154.243 157.23 155.743 154.826 EXPD 0.08869 —– —– -55.587 117.174 119.845 118.888 117.542 TIITLGSIED 5.88639 0.12657 —– -111.94 227.893 229.885 228.599 228.282 TIITLSIED 0.32232 —– —– -118.56 239.123 240.119 239.346 239.318 Figure 3: The empirical distribution and estimated CDF of models using data set 2. Table 11: Descriptive statistics for data set 2 n Minimum Maximum Mean Median Standard deviation Kurtosis Skewness 20 29.71 62.6256 44.4768 42.3049 9.9008 1.9749 0.5360 Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 24 of 27 (a) Estimated PDFs. (b) Boxplot of the data set 2. Figure 4: Plots of the estimated PDF of the models and the Boxplot using data set 2. 5.3. Data set 3 The following dataset is the susceptibility index (SI) of peppermint packets irradiated with gamma rays (6, 8, 10 KGy) or microwave rays (1, 2, 3 min) under choice packet test to Stegobium paniceum L. See [22]. The SI values were: 2.463, 2.53, 2.34, 2.16, 2.10, 2.13, 1.79, 1.86, 1.75, 1.45, 1.35, 1.37, 2.24, 2.32, 2.29, 2.02, 2.09, 2.15, 1.57, 1.48, and 1.47. Table 12: MLEs of the model parameters and the statistics of the L.L, AIC, BIC, CAIC and HQIC for the data set 3 The model The parameters estimate L.L AIC BIC CAIC HQIC α θ λ TIITLGIED 20.5142 5.22388 6.36555 -9.0119 24.0239 27.1575 25.4357 24.704 EGIED 1.70988 5.12732 4.33391 -23.048 52.0966 55.2302 53.5084 52.7767 EXPD 0.51316 —– —– -35.011 72.0211 73.0657 72.2317 72.2478 TIITLGSIED 13.3358 0.33297 —– -26.924 57.8489 59.9379 58.5155 58.3023 TIITLSIED 2.27419 —– —– -29.593 61.1858 62.2299 61.3959 61.4121 Figure 5: The empirical distribution and estimated CDF of models using data set 3. Z. Al-saiary, S. Al-jadaani / Eur. J. Pure Appl. Math, 18 (4) (2025), 5860 25 of 27 Table 13: Descriptive statistics for data set 3 n Minimum Maximum Mean Median Standard deviation Kurtosis Skewness 21 1.35 2.53 1.9487 2.09 0.3787 1.7344 -0.2584 (a) Estimated PDFs. (b) Boxplot of the data set 3. Figure 6: Plots of the estimated PDF of the models and the Boxplot using data set 3. Tables (8,10,12) show that the parameters of the presented models were estimated along with the values of (LL, AIC, BIC, CAIC and HQIC) using three data sets. Thus, we observed from these tables that the TIITL-GIED gives the smallest values compared to other models values for these data sets. Therefore, the TIITL- GIED is the best fitting for this data. Further, Figures (1,3,5) and (2a, 4a, 6a) Show that the TIITL-GIED is the best fitting for this data comparing with the competing models. 6. 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