Adv Syst Sci Appl 2023; 04:60–75 Published online at https://ijassa.ipu.ru. Bayesian Estimation and Prediction from a Mixture of Weibull and Gompertz Distributions Abdulqader Al-Dugin1,2*, Mostafa Mohie El-Din1, Amr Sadek1 1Department of Mathematics, Al-Azhar University, Cairo, Egypt 2Department of Mathematics, Abyan University, Abyan, Yemen Abstract: We study different methods for estimation the parameters of a mixture of Weibull and Gompertz distributions as a lifetime model, based on a complete sample. Maximum likelihood estimation and Bayes estimation under informative and non-informative priors have been obtained using the symmetric squared error (SE) loss function, the asymmetric Linear exponential (LINEX) loss function and general entropy (GE) loss function. Also, we discuss two-sample Bayesian prediction intervals of the proposed model. For the illustration of the developing results, some computation results for the proposed model is presented. Keywords: mixture model; loss function; weibull; Gompertz; maximum likelihood estimation; Bayesian estimation and prediction 1. INTRODUCTION Mixtures models have received great attention from analysts in the recent years due to their important role in life testing and reliability. In many applications, mixture models are used to analyze random duration in possibly heterogeneous populations, statistical analysis and machine learning such as modeling, classification, and survival analysis. Attention has been paid by some authors to the finite mixtures to discuss lifetime distributions, [see, Everitt and Hand (1981),Titterington et al. (1985), Mclachlan and Basford (1988), Lindsay (1995), Mclachlan and Peel (2000)]. Also, mixture distributions have been considered extensively by several researchers using both classical and Bayesian techniques, for example, Shawky and Bakoban (2009), Abu-Zinadah (2010), Erisoglu et al.(2011), Feroze and Aslam (2014), Daniyal and Rajab (2015), Elshahat and Mahmoud (2016) and Mahmoud et al.(2017). The Weibull distribution has been widely used in modeling of lifetime event data; this is due to the variety of shapes of the probability density function (pdf) based on its parameters, Mohie El-Din et al. (2018) presented a new study on progressive-stress accelerated life testing for power generalized Weibull distribution under progressive Type-II censoring. The Weibull distribution has been shown to be useful for lifetime modeling and data analysis in applied engineering sciences, and also the static stress accelerated life test of the generalized Weibull distribution has been studied under step-wise type-II [see, Mohie El-Din et al.(2019), Murthy et al.(2003)]. The Gompertz distribution is used to model human survival and mortality times and actuarial tables. It has many real-life applications, particularly in medical and actuarial studies, and for statistical inference of the Gompertz distribution on the basis of Type- II Hybrid Progressively censored data [see, Mohie El-Din et al. (2017)]. The Gompertz distribution is also used as a survival model in reliability. It has an increased risk rate for the ∗Corresponding author: abdulqader@azhar.edu.eg BAYESIAN ESTIMATION AND PREDICTION 61 life of the systems. Because of its complex shape, it has not received enough attention in the past. However, recently, this distribution has received a lot of attention from demographers and actuaries, and several studies have been presented in this field, including statistical inference and prediction of the Gompertz distribution based on multiply type- I censored data. Mohie El-Din and Abu- Moussa,(2018). The characterization for Gompertz distribution based on general progressively type-II right censored order statistics is done by Mohie El-Din,et al.(2017). Pollard and Valkovics (1992) were the first to deal with the Gompertz distribution thoroughly. However, their results are true only in cases where the initial level of mortality is very close to zero. Willemse and Koppelaar (2000) proposed a new formulation of epistemic elicitation of Gompertz’s law of mortality. If X has the same units and the same rates of variation of two distributions, then a random variable X is said to have a mixture distribution. Let a random variable X is to have a mixture of two components Weibull and Gompertz distribution, with the probability density function (pdf) is given by: f(x) = 2∑ j=1 pjfj(x); ; j = 1, 2, (1.1) where f1(x) = α1θx θ−1e−α1x θ , x > 0, α1 > 0, θ > 0, f2(x) = α2e x−α2(e x−1) , x > 0, α2 > 0. The mixing proportions pj, are such that 0 ≤ pj≤ 1, 2∑ j=1 pj = 1. The corresponding cumulative distribution function(cdf) and reliability function, receptively are given by: F (x) = 2∑ j=1 pjFj(x), j = 1, 2, (1.2) where F1(x) = 1− e−α1x θ , F2(x) = 1− e−α2(e x−1), and R(x) = 2∑ j=1 pjRj(x), j = 1, 2, (1.3) where R1(x) = e−α1x θ , R2(x) = e−α2(e x−1). The objective of this work is to apply the Bayesian procedure to estimate the parameters and obtain two sample prediction bounds for future observations from the proposed model, based on complete sample. The rest of this paper is organized as follows: In Section 2, we obtain maximum likelihood estimators of the proposed parameters. The Bayesian estimation is discussed in Section 3. Bayesian prediction presented in Section 4. Simulation study and real data presented in Section 5 to compare the performance of different estimation methods of the parameters. Finally, conclusions are presented in Section 6. 2. MAXIMUM LIKELIHOOD ESTIMATION In this section, we present maximum likelihood estimators (MLEs) for the unknown parameters α1, α2 and p based on complete sample. Suppose a sample of n units are put on operation in life testing experiment and that the test is terminated if all n items taken Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 62 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK from this population have failed. After the n units have failed each item can be attributed to the appropriate sub-population. Thus if the n units have failed during the interval (0, x(n)); r1 from the first sub-population and r2 from the second sub-population. Let xij denote the failure of the jth unit that belongs to the ith sub-population and xij ≤ x(n); j = 1, 2, ...., ri; i = 1, 2;n = r1 + r2 , where x(n) denotes the failure time of the nth unit. For a two-component mixture model, the likelihood function is given by: L(α1, α2, θ, p |x) = n! [ r1∏ j=1 p1 f 1(x1j)][ r2∏ j=1 p2 f 2(x2j)], (2.4) where f1(x1j) = α1θx θ−1 1j e−α1x θ 1j , x1j > 0, α1 > 0, θ > 0, f2(x2j) = α2e x 2j−α2(e x2j−1), x2j > 0, α2 > 0, then, L(α1, α2, θ, p|x) ∝ r1∏ j=1 p1 α1 θ x θ−1 1j e−α1x θ 1j r2∏ j=2 p2 α2 e x2j−α2(e x2j−1). (2.5) Put p1 = p, p2 = 1− p, and assuming that the parameter θ is known, the likelihood function (2.5) reduces to L(α1, α2, p|x, θ) ∝ 2∏ i=1 (piαi) ri e −α1 r1∑ j=1 x θ 1j e −α2 r2∑ j=1 (ex2j−1) . (2.6) Thus, the log-likelihood function of parameters α1, α2 and p are given by: lnL = lnL(p, α1, α2|x) ∝ 2∑ i=1 {ri ln pi+ri lnαi} − α1 r1∑ j=1 x θ 1j − α2 r2∑ j=1 (ex2j−1). (2.7) Taking derivatives with respect to α1, α2 and p in Equation (2.7), and Equating by zero, the maximum likelihood estimators of the three parameters are obtained as follows α̂1 = r1 r1∑ j=1 xθ ij , α̂2 = r2 r2∑ j=1 (ex2j − 1) , p̂ = r1 r1 + r2 . 3. BAYESIAN ESTIMATION In this section, we derive Bayesian estimators of the parameters α1, α2 and p of the considered model by using various priors based on different symmetric and asymmetric loss functions. 3.1. Loss function In decision theory, the loss criterion is specified in order to obtain the best estimator. Three loss functions are proposed, symmetric squared error loss function (SE) and asymmetric (LINEX and general entropy) loss functions, as follows: Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 63 • Squared error loss function (ES): A simple, and very common loss function is defined by L1(θ̂ , θ) = c (θ̂ − θ)2, where c is constant which is symmetrical in nature and gives equal weight to overestimation as well as underestimation. However, in real applications, estimation of reliability and failure rate functions, an overestimate is more serious than the underestimate. The use of symmetric loss function might be inappropriate as has been recognized by Basu and Ebrahimi (1991). • Linear exponential loss function (LINEX): One of the most commonly used asymmetric loss functions, introduced by Varian (1975) under the assumption that the minimal loss occurs at θ̂ = θ, it can be expressed as L2(θ̂ , θ) ∝ e−q( θ̂ −θ ) − q( θ̂ − θ)− 1, q ̸= 0, where q determines the shape of the loss function. It is used in both overestimation and underestimation, if q > 0 means overestimation and if q < 0, means underestimation but in a situation where q = 0, the LINEX loss function is almost symmetric and approaches squared error loss function. Under the above loss function, the Bayes estimator θ̂LINEX of θ can be obtained as θ̂LINEX = −1 q ln [E (e−qθ|x)], provided that the expected value with respect to the posterior function of θ, E(e−qθ|x) exists and is finite. • General entropy loss function (GE): Another commonly asymmetric loss function is the modified LINEX loss function called a general entropy loss function proposed by Calabria and Pulcini (1996). L3(θ̂, θ) ∝ ( θ̂ θ )h − h ln ( θ̂ θ ) − 1, h ̸= 0, which has a minimum at θ̂ = θ. Also, this loss function used by several authors, in the original form having the shape parameter h = 1, for h > 0, a positive error has a more effect than a negative error. In this case, the Bayes estimate of θ is given by: θ̂GE = [ E ( θ−h|x )]− 1 h , provided that the expected value with respect to the posterior function of θ, E ( θ−h|x ) exists and is finite. 3.2. The posterior distribution under the informative prior Assume the prior distribution of the parameters α1, α2 and p are α1 ∼ Γ(a1, b1) , α2 ∼ Γ(a2, b2), and p ∼ β(c, d) for the mixing parameter, pi; i = 1, 2 where p1 = p and p2 = 1− p. Assuming, now, the independence of parameters, the joint distribution prior for α1, α2 and p is: π(α1, α2 , p)= π1(α1)π2(α2)π3(p), (3.8) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 64 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK where πi(αi) = baii Γ(ai) αai−1 i e−biαi , αi > 0, ai, bi > 0, i = 1, 2, π3(p) = Γ(c+ d) Γ(c)Γ(d) pc−1(1− p)d−1, 0 < p < 1, c, d > 0. Then, the joint prior distribution of α1, α2 and p can be written as follows π (α1, α2 , p) ∝ [ 2∏ i=1 α ai−1 i e−biαi ] × p c−1 ( 1− p) d−1. (3.9) It follows from (2.6) and (3.9), that the joint posterior density function of α1, α2 and p is given by g (α1, α2, p|x)= k−1 1 pr1+c−1 (1− p)r2+d−1 α r1+a1−1 1 α r2+a2−1 2 e−α1ϕ1 e −α2ϕ2 , (3.10) where, k1 is the normalizing constant given by k1 = β (r1+c , r2+d) Γ (r1+a1) Γ (r2+a2) (ϕ1) r1+a1 (ϕ2) r2+a2 , with ϕ1 = b1+ ∑r1 j=1x θ 1j , and ϕ2 = b2+ ∑r2 j=1(e x2j−1). 3.2.1. Bayes estimator under squared error loss function (SE) The Bayes estimators of α1, α2 and p based on the squared error loss function are given by: α̂1SE = k−1 1 β(r1+c , r2+d) Γ(r1+a1+1) Γ (r2+a2) (ϕ1) r1+a1+1 (ϕ2)r2+a2 , α̂2 SE = k−1 1 β(r1+c , r2+d) Γ (r1+a1) Γ(r2+a2+1) (ϕ1)r1+a1(ϕ2)r2+a2+1 , p̂ SE = k−1 1 β(r1+c+ 1 , r2+d ) Γ (r1+a1) Γ (r2+a2) (ϕ1)r1+a1(ϕ2)r2+a2 . 3.2.2. Bayes estimator under Linex loss function (LINEX) The Bayes estimators of α1, α2 and p based on the LINEX loss function are given by: α̂1LINEX = −1 q ln [ k−1 1 β(r1+c , r2+d ) Γ (r1+a1) Γ (r2+a2) (ϕ1+q)r1+a1(ϕ2)r2+a2 ] , α̂2LINEX = −1 q ln [ k−1 1 β (r1+c , r2+d) Γ (r1+a1) Γ (r2+a2) (ϕ1)r1+a1(ϕ2+q)r2+a2 ] , p̂ LINEX = −1 q ln [ k−1 1 Γ (r1+a1) Γ (r2 + a2) (ϕ1)r1+a1(ϕ2)r2+a2 ∞∑ k=0 (−q)k k! β(r1+c+ k , r2+d) ] . Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 65 3.2.3. Bayes estimator under general entropy loss function (GE) The Bayes estimator of α1, α2 and p based on general entropy loss function are given by: α̂1GE = [ k−1 1 β (r1+c , r2+d) Γ(r1+a1−h )Γ (r2+a2) (ϕ1)r1+a1−h(ϕ2)r2+a2 ]− 1 h , α̂2GE = [ k−1 1 β (r1+c, r2+d) Γ (r1+a1) Γ(r2+a2−h) (ϕ1) r1+a1 (ϕ2) r2+a2−h ]− 1 h , p̂GE= [ k−1 1 β(r1+c− h , r2+d) Γ (r1+a1) Γ (r2+a2) (ϕ1)r1+a1(ϕ2)r2+a2 ]− 1 h . 3.3. The posterior distribution under the non-informative prior Jeffery’s prior is a formal rule for obtaining a non-informative prior. The non-informative priors are recommended when there is no formal prior information about the parameters. This is defined as the distribution of the parameters proportional to the square root of the determinants of the Fisher information matrix. The prior distribution for the mixing parameter p, i.e., p ∼ uniform(0,1). Assuming independence, the joint prior distribution is given by: πi(αi) ∝ 1 ai , αi > 0, i = 1, 2, π3(p) = 1, 0 < p < 1. (3.11) Then, the joint prior distribution of α1, α2 and p can be written as follows π (α1, α2, p)∝ 1 α1α2 , α1, α2> 0 , 0 < p < 1. (3.12) It follows from ( 2.6) and ( 3.12) that, the joint posterior density function of α1, α2 and p is given by g (p, α1, α2|x) ∝ k−1 2 pr1 (1− p)r2 αr1−1 1 αr2−1 2 e−α1ϕ ∗ 1 e−α2ϕ ∗ 2 , (3.13) where, k2 is the normalizing constant given by k2 = β(r1+1 , r2+1 ) Γ (r1) Γ (r2) (ϕ∗ 1) r1 (ϕ∗ 2) r2 , with ϕ∗ 1= ∑r1 j=1x θ 1j and ϕ∗ 2= ∑r2 j=1(e x2j−1). 3.3.1. Bayes estimator under squared error loss function (SE) The Bayes estimators of α1, α2 and p based on the squared error loss function are given by: α̂1SEN = k−1 2 β(r1+1, r2+1) Γ(r1+1)Γ (r2) (ϕ∗ 1) r1+1 (ϕ∗ 2) r2 , α̂2 SEN = k−1 2 β(r1+1 , r2+1) Γ (r1) Γ (r2+1) (ϕ∗ 1) r1 (ϕ∗ 2) r2+1 , p̂SEN = k−1 2 β(r1+2 , r2+1) Γ (r1) Γ (r2) (ϕ∗ 1) r1 (ϕ∗ 2) r2 . 3.3.2. Bayes estimator under Linex loss function (LINEX) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 66 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK The Bayes estimators of α1, α2 and p based on the LINEX loss function are given by: α̂1LINEXN = −1 q ln [ k−1 2 β (r1+1, r2+1) Γ(r1)Γ (r2) (ϕ∗ 1+q)r1 (ϕ∗ 2) r2 ] , α̂2LINEXN = −1 q ln [ k−1 2 β (r1+1 , r2+1) Γ (r1) Γ(r2) (ϕ∗ 1) r1 (ϕ∗ 2+q)r2 ] , p̂LINEXN= −1 q ln [ k−1 2 Γ(r1)Γ(r2) (ϕ∗ 1) r1(ϕ∗ 2) r2 ∞∑ k=0 (−q)k k! β(r1+k + 1 , r2+1) ] . 3.3.3. Bayes estimator under general entropy loss function (GE) The Bayes estimators of α1, α2 and p based on general entropy loss function are given by: α̂1GEN = [ k−1 2 β(r1+1, r2+1) Γ(r1−h)Γ(r2) (ϕ∗ 1) r1−h(ϕ∗ 2) r2 ]− 1 h , α̂2GEN = [ k−1 2 β(r1+1 , r2+1) Γ(r1)Γ(r2−h) (ϕ∗ 1) r1(ϕ∗ 2) r2−h ]− 1 h , p̂GEN = [ k−1 2 β(r1 − h+ 1, r2 + 1) Γ(r1)Γ(r2) (ϕ∗ 1) r1(ϕ∗ 2) r2 ]− 1 h . 4. BAYESIAN PREDICTION In this section, the Bayesian two-sample prediction of a future order statistics sample considered on the basis of observed data used in the informative and non-informative prior. A random sample of size m of future observation, independent sample of size n, is drawn from the same population with Eq(1.1) . Therefore Ys represents the sth ordered statistic in the future sample of size m, 1 ⩽ s ⩽ m. The sth order statistic in a sample of size m represents the life length of a (m− s+ 1) out of m system. The distribution function of Ys , the ordered future sample is given by (See, Arnold et al. (1992) and Jaheen (2003)), FYs (ys|α1, α2, p) = m∑ l=s ( m l ) [FX(ys|α1, α2, p)] l [1− FX (ys|α1, α2, p) ] m−l = m∑ l=s l∑ j1=0 ( m l )( l j1 ) (−1)j1 [R(ys)] m−l+j1 , (4.14) where FX (ys|α1, α2, p) = 1−R(ys) is the distribution function of the mixture model and R(ys) is the reliability function of the mixture model after replacing x by ys. Using the binomial expansion for [R(ys)] m−l+j1 as follows: [R(ys)] m−l+j1= [ p1 e −α1y θ s+p2 e −α2(e ys−1) ]m−l+j1 , = m−l+j1∑ j2=0 ( m− l + j1 j2 ) pδ11 pj22 e−α1δ1y θ s e−α2j2(e ys−1). (4.15) Therefore, we get Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 67 FYs (ys|α1, α2, p) = m∑ l=s l∑ j1=0 m−L+j1∑ j2=0 ( m l )( l j1 )( m− l + j1 j2 ) (−1)j1 pδ11 pj22 (e−α1y θ s )δ1 (e−α2(e ys−1))j2 , (4.16) where δ1 = m− l + j1 − j2. f ∗(ys|x) = ∫ 1 0 ∫ ∞ 0 ∫ ∞ 0 f (ys|α1, α2, p) g (α1, α2, p|x) dα1dα2dp, (4.17) where g (α1, α2, p|x) is the joint posterior density for parameters α1, α2 and p and f (ys|α1, α2, p) is the pdf of sth component in a future sample. Therefore, Bayesian prediction density of Ys for a given value v, can be obtained as: P [Ys ≥ v|x] = ∞∫ v f ∗(ys|x)dys =1− ∫ 1 0 ∫ ∞ 0 ∫ ∞ 0 FYs (v|α1, α2, p) g (α1, α2, p|x) dα1dα2dp, (4.18) Substitution of (3.9) and (4.16) in (4.18), we get Bayes predictive distribution bounds with value v for Ys in case of informative as P [Ys ≥ v |x] = 1− k−1 1 ∑ B β (δ1+δ2, δ3) Γ (r1+a1) Γ (r2+a2) (ϕ∗∗ 1 )r1+a1 (ϕ∗∗ 2 )r2+a2 . where δ2 = r1 + c, δ3 = r2 + d+ j2. Substitution of (3.11) and (4.16) in (4.18), we get Bayes predictive distribution bounds with value v for Ys in case of non- informative as p [Ys ≥ v |x] = 1− k−1 2 ∑ B β (δ∗1+1 , δ∗2+1) Γ (r1) Γ (r2) (ϕ∗∗∗ 1 )r1 (ϕ∗∗∗ 2 )r2 . where∑ = m∑ l=s l∑ j1=0 m−l+j1∑ j2 , B = ( m l )( l j1 )( m−l+j1 j2 ) (−1)j1 δ∗1 = δ1 + r1 , δ∗2 = j2 + r2 ϕ∗∗ 1 = δ1v θ + ϕ1 , ϕ∗∗ 2 = j2(e v − 1) + ϕ2 ϕ∗∗∗ 1 = δ1v θ + ϕ∗ 1 , ϕ∗∗∗ 1 = j2(e v − 1) + ϕ∗ 2 A 100 γ% prediction interval for Ys is given by P [L(x) < Ys < U(x)] = γ, where L(x) and U(x) are obtained respectively by solving the following two equations: P [Ys > L(x)] = 1 + γ 2 and P [Ys > U(x)] = 1− γ 2 . (4.19) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 68 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK 5. SIMULATION STUDY AND REAL DATA In this section, we present some simulation results to compare the performance of various estimates by using the Monte Carlo simulation method from the mixture of Weibull and Gompertz distributions. The following steps were considered: 1. For the parameter we have considered (α1, α2, p) = (0.2, 0.1, 0.7) along with θ is known and θ = 0.61. The values of q and h are (0.5, -0.5) for LINEX and general entropy loss functions. The method of choosing the hyper parameters values introduced in Ahmadi et al.(2020). Let (a1 = 0.03, a2 = 0.04, b1 = 0.2, b2 = 0.35, c = 96.23, and d = 41.167) for informative prior. In case of non-informative prior, we take (a1 = a2 = b1 = b2 = 0; c = d = 1). In all these cases samples of size n = 20, 40, 60, 80, 100 and 150, are generated. 2. Generate a uniform random number u from the interval(0,1). 3. If u ⩽ p the observation has been randomly taken from first sub-population and if u > p then the observation have been taken from the second sub-population. 4. The obtained results for maximum likelihood estimates and Bayes estimates are calculated. 5. Bayesian prediction for the future observations of Ys are obtained by solving numerically, Equations (4.6) with γ = 0.95. 6. The above steps are repeated 1000 times, and the average of the estimates are calculated and presented in the Tables (5.1)-(5.3). Also the average lower and average upper interval of Ys when s = m 2 is even or m+ 1 2 is odd for different sample size n and different future sample size m, simulated coverage probability and average interval lengths are obtained in Tables (5.4)-(5.5). All results were obtained by using Mathematica 10. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 69 Table 5.1. Average estimates and corresponding MSE of the parameter α1 = 0.2 based on informative and non- informative Prior n MLE Bayes SE LINEX GE informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.217318 0.217035 0.21770 0.217996 0.216086 0.218285 0.216363 0.213051 0.205008 0.21332 0.205247 (0.00451) (0.00446) (0.00475) (0.00458) (0.00435) (0.00463) (0.0044) (0.00418) (0.00372) (0.00423) (0.00376) 40 0.209797 0.209695 0.21531 0.210106 0.209287 0.210209 0.209388 0.207823 0.204061 0.207922 0.204154 (0.00176) (0.00175) (0.00273) (0.00177) (0.00173) (0.00178) (0.0017) (0.00169) (0.00158) (0.00169) (0.00159) 60 0.205851 0.205791 0.20781 0.206052 0.205531 0.206113 0.205591 0.204558 0.202084 0.204617 0.202141 (0.00107) (0.00107) (0.00116) (0.00108) (0.00106) (0.00108) (0.0011) (0.00105) (0.00101) (0.00105) (0.00101) 80 0.202925 0.202884 0.20450 0.203072 0.202696 0.203113 0.202737 0.2019757 0.200154 0.202016 0.200194 (0.00076) (0.00076) (0.00098) (0.00077) (0.00076) (0.00077) (0.0008) (0.00075) (0.00073) (0.00075) (0.00074) 100 0.202708 0.202675 0.20293 0.202825 0.202526 0.202858 0.202558 0.201949 0.200494 0.201981 0.200525 (0.00063) (0.00062) (0.00067) (0.00063) (0.00062) (0.00063) (0.0006) (0.00062) (0.0006) (0.00062) (0.00061) 150 0.201981 0.200088 0.20288 0.202024 0.20209 0.202079 0.201883 0.200249 0.200195 0.198195 0.200575 (0.00041) (0.00078) (0.00052) (0.00055) (0.00042) (0.00041) (0.0004) (0.00064) (0.00044) (0.00106) (0.0004) Table 5.2. Average estimates and corresponding MSE of the parameter α2 = 0.1 based on informative and non- informative Prior n MLE Bayes SE LINEX GE informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.122528 0.121892 0.12360 0.123059 0.120817 0.123818 0.121374 0.116016 0.103891 0.116521 0.104117 (0.00532) (0.004878) (0.00641) (0.00532) (0.00454) (0.00594) (0.00488) (0.00401) (0.00273) (0.00432) (0.00287) 40 0.111112 0.11105 0.11245 0.111384 0.110721 0.111449 0.11078 0.10856 0.103519 0.10861 0.103544 (0.00183) (0.001781) (0.00289) (0.00183) (0.00173) (0.00188) (0.00178) (0.00163) (0.00139) (0.00168) (0.00142) 60 0.108024 0.108021 0.10953 0.108201 0.107841 0.108205 0.107844 0.106479 0.103373 0.106479 0.103365 (0.00079) (0.00079) (0.00095) (0.0008) (0.00078) (0.0008) (0.00079) (0.00074) (0.00067) (0.00075) (0.00067) 80 0.103682 0.103692 0.10633 0.103813 0.103572 0.103803 0.103562 0.102581 0.100348 0.10257 0.100332 (0.00043) (0.000433) (0.00079) (0.00044) (0.00043) (0.00044) (0.00043) (0.00042) (0.00039) (0.00042) (0.00039) 100 0.103475 0.103483 0.10521 0.103578 0.103389 0.103569 0.10338 0.102607 0.100849 0.102598 0.100837 (0.00041) (0.000411) (0.00052) (0.00041) (0.00041) (0.00041) (0.00041) (0.0004) (0.00038) (0.0004) (0.00038) 150 0.101673 0.101554 0.10172 0.101613 0.101495 0.101733 0.101614 0.100986 0.0998459 0.101101 0.099953 (0.00027) (0.000266) (0.00029) (0.00027) (0.00027) (0.00027) (0.00027) (0.00026) (0.00026) (0.00027) (0.00026) Table 5.3. Average estimates and corresponding MSE of the parameter p = 0.7 based on informative and non- informative Prior n MLE Bayes SE LINEX GE informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.69115 0.699258 0.69973 0.699589 0.6989326 0.676418 0.671834 0.69878 0.367819 0.670447 0.662696 (0.01073) (0.000173) (0.00947) (0.00017) (0.00017) (0.0093) (0.00967) (0.00017) (0.00018) (0.009988) (0.01083) 40 0.7034 0.701061 0.70132 0.701354 0.700767 0.694919 0.692503 0.70064 0.699792 0.691894 0.68816 (0.0052) (0.00026) (0.00474) (0.00027) (0.00026) (0.00471) (0.0048) (0.00027) (0.00027) (0.00483) (0.00502) 60 0.697767 0.699586 0.69986 0.69985 0.699321 0.692219 0.690552 0.699205 0.698441 0.690144 0.687615 (0.00346) (0.00032) (0.00331) (0.00032) (0.00032) (0.00329) (0.0033) (0.00032) (0.00032) (0.00336) (0.00347) 80 0.700113 0.700282 0.70053 0.700521 0.700042 0.695861 0.6946 0.699937 0.699244 0.694303 0.692422 (0.00272) (0.00037) (0.00261) (0.00037) (0.00037) (0.0026) (0.0026) (0.00037) (0.00037) (0.00264) (0.0027) 100 0.69908 0.699832 0.69991 0.700052 0.699612 0.695685 0.694666 0.699516 0.698881 0.69443 0.692921 (0.00203) (0.00036) (0.00197) (0.00036) (0.00036) (0.00197) (0.002) (0.00036) (0.00036) (0.00199) (0.00203) 150 0.700313 0.699868 0.69998 0.699949 0.699786 0.69802 0.697335 0.699506 0.698982 0.697181 0.69618 (0.00136) (0.00035) (0.00132) (0.00035) (0.00035) (0.00132) (0.0013) (0.00035) (0.00035) (0.00133) (0.00135) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 70 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK Table 5.4. The 95% Bayesian prediction bounds, length of Bayesian prediction and their simulated coverage probability for Ys based informative prior (n,m) Y1 Y4 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,8) (0.000303,0.4654) 0.465106 0.934 (0.103358,1.96123) 1.85788 0.934 (0.620589,34.0201) 33.3995 0.977 (20,8) (0.000275,0.45017) 0.449896 0.936 (0.106456,1.90021) 1.79375 0.935 (0.760848,32.4311) 31.6703 0.967 (30,8) (0.000265,0.44291) 0.442654 0.945 (0.109328,1.86521) 1.75588 0.945 (0.463508,30.5226) 30.0591 0.981 (50,8) (n,m) Y1 Y5 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,10) (0.000216,0.36387) 0.363661 0.946 (0.130549,1.82091) 1.69036 0.933 (0.879757,39.4524) 38.5727 0.958 (20,10) (0.0002006,0.3515) 0.351314 0.941 (0.137363,1.78166) 1.6443 0.944 1.05329,34.2686) 33.2153 0.957 (30,10) (0.000190,0.34323) 0.343049 0.951 (0.141384,1.72623) 1.58484 0.936 (0.936194,32.7176) 31.7814 0.963 (50,10) (0.000174,0.33177) 0.331604 0.946 (0.145217,1.69538) 1.55017 0.937 (0.779346,34.6254) 33.8461 0.983 (n,m) Y1 Y6 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,12) (0.000165,0.29767) 0.297512 0.935 (0.1511,1.69131) 1.54021 0.929 (1.06404,39.1433) 38.0793 0.968 (20,12) (0.000154,0.28254) 0.282393 0.945 (0.150129,1.61944) 1.46931 0.936 (0.900551,38.0184) 37.1179 0.965 (30,12) (0.000139,0.27243) 0.272296 0.954 (0.165389,1.63105) 1.46566 0.946 (0.861137,34.6709) 33.8098 0.978 (50,12) (0.000129,0.26469) 0.264562 0.947 (0.169259,1.56466) 1.3954 0.947 (0.949135,33.8375) 32.8884 0.978 (n,m) Y1 Y7 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,13) (0.000152,0.27705) 0.277058 0.938 (0.193216,1.9467) 1.75349 0.936 (0.964702,42.2606) 41.2959 0.973 (20,13) (0.000141,0.26195) 0.26181 0.941 (0.208467,2.03536) 1.82689 0.935 (1.17018,39.2434) 38.0732 0.976 (30,13) (0.0001234,0.24464) 0.244524 0.945 (0.210109,1.90317) 1.69306 0.931 (1.17348,35.6211) 34.4476 0.966 (50,13) (0.0001143,0.23821) 0.238096 0.957 (0.211075,1.81535) 1.60427 0.958 (1.18585,33.1661) 31.9803 0.982 Table 5.5. The 95% Bayesian prediction bounds,length of Bayesian prediction and their simulated coverage probability for Ys based non - informative prior (n,m) Y1 Y4 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,8) (0.000248,0.49451) 0.49427 0.953 (0.111943,2.86765) 2.75571 0.948 (0.779858,33.9345) 33.1547 0.955 (20,8) (0.0002421,0.46972) 0.469481 0.951 (0.102928,2.46754) 2.36461 0.949 (0.558534,36.6563) 36.0978 0.988 (30,8) (0.000240,0.46191) 0.461669 0.963 (0.106012,2.21552) 2.10951 0.961 (0.537214,32.6924) 32.1552 0.979 (50,8) (0.000235,0.45122) 0.450991 0.948 (0.108414,2.01425) 1.90583 0.952 (0.183484,30.7059) 30.5224 0.988 (n,m) Y1 Y5 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,10) (0.000187,0.38934) 0.389161 0.954 (0.147564,2.78801) 2.64045 0.948 (1.14207,45.6324) 44.4903 0.949 (20,10) (0.000177,0.36678) 0.366609 0.951 (0.141181,2.34187) 2.20069 0.947 (1.11316,37.6697) 36.5566 0.961 (30,10) (0.000165,0.34543) 0.345268 0.957 (0.1373962.01525) 1.87786 0.953 (1.05137,32.9594) 31.908 0.968 (50,10) (0.0001621,0.33803) 0.337875 0.947 (0.143411,1.86337) 1.71996 0.953 (1.06816,39.8492) 38.7811 0.973 (n,m) Y1 Y6 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,12) (0.0001403,0.3082) 0.30812 0.961 (0.16975,2.64656) 2.47681 0.923 (1.31578,45.0772) 43.7614 0.953 (20,12) (0.0001340,0.291961) 0.291827 0.952 (0.173968,2.29415) 2.12018 0.939 (1.17472,40.9827) 39.808 0.967 (30,12) (0.0001238,0.275178) 0.275055 0.947 (0.179458,2.0297) 1.85024 0.961 (1.35481,38.1587) 36.8039 0.968 (50,12) (0.0001214,0.265905) 0.265784 0.962 (0.171414,1.74186) 1.57044 0.959 (1.33663,33.6541) 32.3175 0.974 (n,m) Y1 Y7 Ym (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (Lower, Upper) Length Coverage (15,13) (0.0001372,0.28479) 0.284626 0.941 (0.205341,2.78688) 2.58154 0.939 (1.01779,49.069) 48.0512 0.983 (20,13) (0.000191,0.266334) 0.266213 0.949 (0.206602,2.48416) 2.27756 0.956 (1.37849,43.4506) 42.0721 0.967 (30,13) (0.0001098,0.24637) 0.246269 0.956 (0.217119,2.22986) 2.01274 0.961 (1.1813,37.8901) 36.7088 0.976 (50,13) (0.000125,0.23432) 0.234541 0.954 (0.224957,1.99452) 1.76956 0.958 (1.4746,33.8247) 32.3501 0.958 Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 71 5.1. Numerical Example Now, we presents a numerical example to illustrate the methodology for the proposed estimates based on real data. Consider the following data set is from Kotz and Johhnson (1983) and represents the survival times (in years) after diagnosis of 43 patient with certain kind of leukemia. The data has been classified into two sets using probabilistic mixing weights for p=0.7, which produced r1 = 30 and r2 = 13,, as follows: population-I population-II 0.019, 0.159, 0.636, 0.748, 1.175, 1.206, 1.282, 1.356, 1.362, 1.458, 1.564, 1.586, 1.592, 1.781, 1.923, 1.959, 2.134, 2.548, 2.652, 2.951, 3.038, 3.655, 3.754, 4.690, 4.888, 5.143, 5.167, 5.603, 6.192, 6.874 6.655, 0.129, 0.485, 2.466, 0.203, 4.203, 2.413, 1.219, 1.219, 0.781, 0.869, 3.6, 5.633 The estimated and prediction results are presented in Tables [(5.6)-(5.11)]. Table 5.6. Average estimates corresponding to real data set in case informative Prior Parameter MLE Bayes Loss Function SE LINEX GE q=0.5 q=-0.5 q=1 q=-1 h=0.5 h=-0.5 h=1 h=-1 α1 0.241776 0.241112 0.240629 0.241597 0.240148 0.242086 0.235093 0.239111 0.233075 0.241112 α2 0.010971 0.0109709 0.0109686 0.0109733 0.0109663 0.0109756 0.0103402 0.010762 0.0101271 0.0109709 p 0.697674 0.65625 0.655097 0.657399 0.653939 0.658542 0.650806 0.654462 0.648936 0.65625 Table 5.7. Average estimates corresponding to real data set in case non- informative Prior Parameter MLE Bayes Loss Function SE LINEX GE q=0.5 q=-0.5 q=1 q=-1 h=0.5 h=-0.5 h=1 h=-1 α1 0.241776 0.241776 0.24129 0.242264 0.240807 0.242755 0.23574 0.239769 0.233716 0.241776 α2 0.010971 0.0109713 0.010969 0.0109736 0.0109667 0.010976 0.0103405 0.0107624 0.0101274 0.0109713 p 0.697674 0.688889 0.687721 0.690051 0.686547 0.691206 0.683628 0.687163 0.681818 0.688889 Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 72 A. AL-DUGIN, M. M. MOHIE EL-DIN, A. SADEK Table 5.8. Bayesian prediction bounds Ys, length of the Bayesian prediction corresponding 90% in case informative prior for the real data set (n,m) Y1 Y4 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,8) (0.0897561,2.00536) 1.9156039 (1.23083,4.29157) 3.06074 (4.01878,9.16364) 5.14486 (n,m) Y1 Y5 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,10) (0.0759191,1.69281) 1.6168909 (1.38585,4.2249) 2.83905 (4.29473,9.51146) 5.21673 (n,m) Y1 Y6 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,12) (0.0662061,1.4729) 1.4066939 (1.50481,4.17095) 2.66614 (4.49234,9.79406) 5.30172 (n,m) Y1 Y7 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,13) (0.0623424,1.38559) 1.3232476 (1.72943,4.32339) 2.59396 (4.57268,9.9177) 5.34502 Table 5.9. Bayesian prediction bounds Ys, length of the Bayesian prediction corresponding 90% in case non-informative prior for the real data set (n,m) Y1 Y4 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,8) (0.0870625,1.93105) 1.8439875 (1.18344,4.27051) 3.08707 (3.97367,9.4282) 5.45453 (n,m) Y1 Y5 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,10) (0.0736783,1.62945) 1.5557717 (1.33145,4.19961) 2.86816 (4.27084,9.78129) 5.51045 (n,m) Y1 Y6 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,12) (0.0642801,1.41804) 1.3537599 (1.44492,4.14233) 2.69741 (4.4837,10.0682) 5.5845 (n,m) Y1 Y7 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,13) (0.0605408,1.33419) 1.2736492 (1.65938,4.30482) 2.64544 (4.57015,,10.1938) 5.62365 Table 5.10. Bayesian prediction bounds Ys, length of the Bayesian prediction corresponding 95% in case informative prior for the real data set (n,m) Y1 Y4 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,8) (0.0528,2.36052) 2.30772 (1.03033,4.54519) 3.51486 (3.69989,10.2809) 6.58101 (n,m) Y1 Y5 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,10) (0.0446395,1.99962) 1.9549805 (1.18738,4.46186) 3.27448 (4.01896,10.6249) 6.60594 (n,m) Y1 Y6 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,12) (0.0389124,1.74192) 1.7030076 (1.31001,4.39521) 3.0852 (4.24416,10.9047) 6.66054 (n,m) Y1 Y7 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,13) (0.0366346,1.63903) 1.6023954 (1.52286,4.53485) 3.01199 (4.33457,11.0273) 6.69273 Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) BAYESIAN ESTIMATION AND PREDICTION 73 Table 5.11. Bayesian prediction bounds Ys, length of the Bayesian prediction corresponding 95% in case non-informative prior for the real data set (n,m) Y1 Y4 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,8) (0.0513029,2.27814) 2.2268371 (0.991506,4.54152) 3.550014 (3.63099,10.5643) 6.93331 (n,m) Y1 Y5 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,10) (0.0434,1.92644) 1.88304 (1.14161,4.45264) 3.31103 (3.97304,10.9139) 6.94086 (n,m) Y1 Y6 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,12) (0.0378516,1.67744) 1.6395884 (1.25864,4.38166) 3.12302 (4.21589,11.1984) 6.98251 (n,m) Y1 Y7 Ym (Lower, Upper) Length (Lower, Upper) Length (Lower, Upper) Length (63,13) (0.0356443,1.57834) 1.5426957 (1.46169,4.53063) 3.06894 (4.31345,11.323) 7.00955 6. CONCLUSION In this paper, we have addressed the estimation and prediction problems of the mixture of Weibull and Gompertz distributions. Different estimators of the parameters are obtained using maximum likelihood and Bayesian methods, under the informative and non-informative prior distributions, we conclude. 1. The Bayes estimates perform better under informative prior that non-informative prior for all different loss functions. 2. The mean squared error of maximum likelihood estimates and Bayesian estimates of the proposed parameters decrease as the sample size increases. 3. The estimates of α1 and α2 in the case of asymmetric loss function is better, at the positive values of q and h than with negative values. 4. Tables (5.4)and (5.5) show that the lengths of the Bayesian prediction intervals lengths decrease as the sample size increases, and that the Bayesian simulated coverage probability of Ys one when to the confidence level. The lengths of the Bayesian prediction intervals increases as s increases. 5. 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Adv Syst Sci Appl (2023) Introduction Maximum Likelihood Estimation Bayesian Estimation Loss function The posterior distribution under the informative prior Bayes estimator under squared error loss function (SE) Bayes estimator under Linex loss function (LINEX) Bayes estimator under general entropy loss function (GE) The posterior distribution under the non-informative prior Bayes estimator under squared error loss function (SE) Bayes estimator under Linex loss function (LINEX) Bayes estimator under general entropy loss function (GE) Bayesian Prediction Simulation Study and Real Data Numerical Example Conclusion