# The Weibull Claim Model: Bivariate Extension, Bayesian, and Maximum Likelihood Estimations

Authors: Walid Emam; Yusra Tashkandy
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

Using a class of claim distributions, we introduce the Weibull claim distribution, which is a new extension of the Weibull distribution with three parameters. The maximum likelihood estimation method is used to estimate the three unknown parameters, and the asymptotic confidence intervals and bootstrap confidence intervals are constructed. In addition, we obtained the Bayesian estimates of the unknown parameters of the Weibull claim distribution under the squared error and linear exponential function (LINEX) and the general entropy loss function. Since the Bayes estimators cannot be obtained in closed form, we compute the approximate Bayes estimates via the Markov Chain Monte Carlo (MCMC) procedure. By analyzing the two data sets, the applicability and capabilities of the Weibull claim model are illustrated. The fatigue life of a particular type of Kevlar epoxy strand subjected to a fixed continuous load at a pressure level of 90% until the strand fails data set was analyzed.


---

## Body

## 1. Introduction

The use of statistical distributions to model life phenomena has attracted considerable research interest. Recent articles have demonstrated the potential of statistical distributions in modelling life data. The Weibull distribution with two parameters is a well-known model that can be effectively used for data modelling in lifetime analysis. The Weibull distribution was introduced by Frechet [1] and first applied by Rosin and Rammler [2] to describe the distribution of a particle size. The Weibull distribution has many applications in most fields. Let X be a random variable (R.V.) that follows the two-parameter Weibull distribution λ,γ, then its cumulative distribution function (CDF), denoted by Fx;λ,γ, is given by(1)Fx;λ,γ=1−exp−xλγ,x≥0;λ,γ>0.The corresponding probability density function (PDF), survival function (SF), and hazard rate function (HRF) of the Weibull R.V. are given, respectively, by(2)fx;λ,γ=γλxλγ−1exp−xλγ,x>0,Sx;λ,γ=exp−xλγ,x>0,and(3)hx;λ,γ=γλxλγ−1,x>0.In this article, we focus on modelling a new three-parameter modification of the Weibull distribution, the Weibull claim distribution. In 1997, Marshall and Olkin [3] developed a new family by adding a shape parameter to the basic distribution, which many researchers used to find and study new distributions. The Weibull claim distribution is introduced using the class of claim distributions introduced by Ahmad et al. [4] based on the Marshall and Olkin mechanism. The CDF and the PDF of a class of claim distributions are, respectively, given by(4)Gx;σ,λ,γ=σFx;λ,γ21−1−σFx;λ,γ,x∈ℝ,and(5)gx;σ,λ,γ=σfx;λ,γFx;λ,γ2−1−σFx;λ,γ1−1−σFx;λ,γ2,x∈ℝ.In the field of Big Data science and other related fields, the best possible description of real-world phenomena is an important research topic (see [5–8] for details). Recent studies have shown the potential of statistical models in various areas of applied science. The aim of this paper is to deepen this research area of distribution theory and propose a new statistical model that provides a better fit to survival time data.The structure of this paper is as follows. Section2 introduces the Weibull claim distribution. The bivariate extension of the Weibull claim distribution is discussed in Section 3. In Section 4, we first discuss the likelihood estimation of the Weibull claim distribution and the asymptotic and bootstrap confidence intervals based on the observed Fisher information matrix. We then discuss the Bayesian estimation of the unknown parameters under the squared error, LINEX, and the general entropy loss function and perform a Monte Carlo simulation study. In Section 5, simulations and two real data sets are performed to evaluate the efficiency of the proposed model. Section 6 provides a brief conclusion.


## 2. The Weibull Claim Distribution

The CDF and the PDF of the Weibull claim distribution can be constructed directly by substituting the formulas (1)–(3) into (4) and (5), respectively. Now, X is a Weibull claim R.V. if its CDF is given by(6)Gx;σ,λ,γ=σ1−exp−x/λγ21−1−σ1−exp−x/λγ,x∈ℝ.The corresponding PDF is given by(7)gx;σ,λ,γ=σγ/λx/λγ−1exp−x/λγ1−exp−x/λγ2−1−σ1−exp−x/λγ1−1−σ1−exp−x/λγ2,x∈ℝ.The main motivations for using the Weibull claim in practice are as follows:(1)
A very simple and convenient way to modify the Weibull distribution.(2)
The improvement of the functions and flexibility of the Weibull distribution.(3)
The introduction of an extended version of the Weibull distribution with a closed form of the distribution function.(4)
The best fit to data in many sciences and fields.(5)
Another important motivation for the proposed approach is the introduction of new distributions by adding only one additional parameter instead of two or more parameters.


## 3. Bivariate Weibull Claim Distribution

Morgenstern [9] introduced the copula model. Copula models are used to represent the common CDF of the two marginal univariate distributions. If Gxj is the CDF of X1 and X2 and ρ is the dependence measure between X1 and X2, the joint CDF defined by the copula model, denoted by Gx1,x2, is given by(8)Gx1,x2=CρGx1,Gx2.Conway [10] introduced the joint CDF and PDF of the copula model, respectively, by(9)Gx1,x2=Gx1Gx21+ρ1−Gx11−Gx2,−1<ρ<1,and(10)gx1,x2=gx1gx21+ρ1−2Gx11−2Gx2.If the R.V.sX1 and X2 follow the Bivariate Weibull claim distribution, then its CDF is given by(11)Gx1,x2=σ11−exp−x1/λ1γ121−1−σ11−exp−x1/λ1γ1×σ21−exp−x2/λ2γ221−1−σ21−exp−x2/λ2γ2×1+ρ1−σ11−exp−x1/λ1γ121−1−σ11−exp−x1/λ1γ1×1−σ21−exp−x2/λ2γ221−1−σ21−exp−x2/λ2γ2,−1<ρ<1,where x1,x2,λ1,γ1,σ1,λ2,γ2,σ2>0. The corresponding PDF be(12)gx1,x2=σ1γ1/λ1x1/λ1γ1−1exp−x1/λ1γ11−exp−x1/λ1γ12−1−σ11−exp−x1/λ1γ11−1−σ11−exp−x1/λ1γ12×1+ρ1−2σ11−exp−x1/λ1γ121−1−σ11−exp−x1/λ1γ11−2σ21−exp−x2/λ2γ221−1−σ21−exp−x2/λ2γ2×σ2γ2/λ2x2/λ2γ2−1exp−x2/λ2γ21−exp−x2/λ2γ22−1−σ21−exp−x2/λ2γ21−1−σ21−exp−x2/λ2γ22.Figures1 and 2 show different PDF and CDF values of the bivariate Weibull claim model for ρ=1. Figure 3, on the other hand, shows the SF and HF of the bivariate Weibull distribution model.Figure 1
Different plot for the PDF of the Bivariate Weibull claim distribution forσ1=3,λ1=1,γ1=1,σ2=5,λ2=4,γ2=2, and ρ=−0.9 (a) and ρ=−1 (b).
(a)(b)Figure 2
Plots of the PDF (a) and CDF (b) of the Bivariate Weibull claim model forσ1=3,λ1=1,γ1=1,σ2=5,λ2=4, and γ2=2.
(a)(b)Figure 3
Plots of the SF (a) and HF (b) of the Bivariate Weibull claim model forσ1=3,λ1=1,γ1=1,σ2=5,λ2=4,γ2=2, and ρ=1.
(a)(b)


## 4. Parameter Estimation

Many studies were examined in the articles to estimate the three unknown parameters. Letx=x1,x2,…,xn be a sample from the Weibull claim distribution.


### 4.1. Maximum Likelihood Method

The likelihood function corresponding to equation (7) is given by(13)Lσ,λ,γ=n!σγλγn∏i=1nxiγ−1exp−∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,x∈ℝ.The log-likelihood function corresponding to equation (13) is given by(14)ℓ=logn!+nlogσ+nlogγ−nγlogλ+nγ−1∑i=1nlogxi−∑i=1nxiλγ+∑i=1nlog1−exp−xiλγ+∑i=1nlog2−1−σ1−exp−xiλγ−2∑i=1nlog1−1−σ1−exp−xiλγ.The equation in equation (14) can be solved by using Newton–Raphson method. The corresponding partial derivatives are(15)∂ℓ∂λ=γλn+1λ∑i=1nxixiλ−1+γ−∑i=1ne−xi/λγxixi/λ−1+γ1−e−xi/λγ+1−σ∑i=1ne−xi/λγxixi/λ−1+γ2−1−e−xi/λγ1−σ−2∑i=1ne−xi/λγxixi/λ−1+γ1−1−e−xi/λγ1−σ,∂ℓ∂γ=n1γ−logλ+∑i=1nlogxi−∑i=1nlogxiλxiλγ+∑i=1ne−xi/λγlogxi/λxi/λγ1−e−xi/λγ+1−σ2∑i=1ne−xi/λγlogxi/λxi/λγ1−1−e−xi/λγ1−σ−∑i=1ne−xi/λγlogxi/λxi/λγ2−1−e−xi/λγ1−σ,And,(16)∂ℓ∂σ=nσ−2∑i=1n1−e−xi/λγ1−1−e−xi/λγ1−σ+∑i=1n1−e−xi/λγ2−1−e−xi/λγ1−σ.The maximum likelihood estimators for the parameters of the Weibull distributionλ,γ, and σ are the solutions of the above nonlinear equations.Then, we can calculate the asymptotic confidence intervals of the parametersσ,λ, and γ. The covariance matrix of the observed variance for the MLEs of the parameters V^=σi,j,i,j=1,2,3 was assumed as follows:(17)V^=−∂2l∂σ2∂2l∂σ∂λ∂2l∂σ∂γ∂2l∂λ∂σ∂2l∂λ2∂2l∂λ∂γ∂2l∂γ∂σ∂2l∂γ∂λ∂2l∂γ2σ=σ^ML,λ=λ^ML,γ=γ^ML−1.1001−α% two-sided approximate confidence intervals for the parameters σ, λ, and γ are then given by(18)σ^±zα/2Vσ^,λ^±zα/2Vλ^,and(19)γ^±zα/2Vγ^,respectively, where Vσ^, Vλ^, and Vγ^ are the estimated variances of σ^ML, λ^ML, and γ^ML, which are given by the diagonal elements of V^, and zα/2 is the upper α/2 percentile of the standard normal distribution.Next, obtaining the bootstrap C.I. for boot-p for the unknown parametersϕ=σ,λ,γ, we apply the following algorithms (for more details, see [11, 12]).Boot-p interval’s algorithm is as follows:Step 1. Generate x1:n,x2:n,…,xn:n from the Weibull claim distribution and derive an estimate ϕ^ of ϕ.Step 2. Generate another sample x1:n∗,x2:n∗,…,xn:n∗ using ϕ^. Then, derive the updated bootstrap estimate ϕ^∗ from ϕ.Step 3. Repeat step 2 with a given number B of repetitions.Step 4. Using F^x=Pϕ^∗≤x, i.e., the CDF of ϕ^∗, the 1001−t% C.I. of ϕ is given by(20)ϕ^Boot−pα2,ϕ^Boot−p1−α2,whereϕ^Boot−px=F^−1x and x is prefixed.


### 4.2. Bayesian Estimation

Bayesian inference is an appropriate method to work with the full samples of the Weibull claim distribution. Given that Weibull claim distributions are so rare, prior information is very useful. We assume thatσ,λ, and γ are R.V.s following prior DFs: Uniform σ;0,aσ, Uniform λ;0,aλ, and Uniform γ;0,aγ, respectively. The posterior DF of σ,λ,andγ and the data under the uniform priors can take the form as follows:(21)πu∗σ,λ,γ|x∝Lσ,λ,γ|xπ1σ,λ,γ,=n!Juaσaλaγσγλγn∏i=1nxiγ−1exp−∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,where J1 is the normalizing constant. Next, we suppose that σ,λ, and γ are R.V.s that follow the prior DF Gamma σ;a1,b1, Gamma λ;a2,b2, and Gamma γ;a3,b3, respectively, where ai and bi are positive constants and i=1,2,3. The posterior DF of σ,λ,γ and the data under the Gamma priors can take the forms as follows:(22)πg∗σ,λ,γ|x∝Lσ,λ,γ|xπ1σ,λ,γ,=n!σa1−1λa2−1γa3−1Jgσγλγn∏i=1nxiγ−1×exp−b1σ+b2λ+b3γ+∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,where Jg is the marginal probability DF of x.


### 4.3. MCMC Method

In the following algorithm, we use Metropolis Hastings (M-H) procedure with normal DF to simulate samples from the distributions:(1)
Set the initial valuesσ0,λ0, and γ0. Then, simulate sample of size n from Weibull claim distribution, next let l=1.(2)
Simulateσ∗,λ∗, and γ∗. using the proposal distributions Nσl−1,Varσ^, Nλl−1,Varλ^, and Nγl−1,Varγ^.(3)
Obtain the probabilityr=minπ∗σ∗,λ∗,γ∗/π∗σl−1,λl−1,γl−1,1.(4)
SimulateU from Uniform (0, 1).(5)
IfU<r, then σl,λl,γl=σ∗,λ∗,γ∗. If U≥r, then σl,λl,γl=σl−1,λl−1,γl−1.(6)
Setl=l+1.(7)
Iterate Steps 2–6,M repetitions, and get σl,λl, and γl for l=1,…,M.In order to conduct a Bayesian analysis, usually quadratic loss function is considered. A very popular quadratic loss is the squared error loss function given byLSEϑ,ϑ^=ϑ−ϑ^2, where ϑ^ is an estimate of the unknown parameter ϑ against SE loss function. By using the generated random samples from the above Gibbs sampling technique and for N is the n burn, then the Bayes estimator of ϑ, say ϑ^SE, can be obtained as(23)ϑ^SE=Eϑϑ|x=1M−N∑l=N+1Mϑl.The second loss function is the LINEX loss function, given by(24)LLEϑ,ϑ^=expρϑ−ϑ^−ρϑ−ϑ^−1,ρ≠0.The approximate Bayes estimate ofϑ=σ,λ,γ under LE loss function based on the Gibbs sampling technique becomes(25)ϑ^LE=−1ρlogEϑexp−ρϑ|x=−1ρlog∑l=N+1Mexp−ρϑlM−N.Finally, the general entropy (GE) loss function is given by(26)LGEϑ,ϑ^=ϑ^ϑε−εlogϑ^ϑ−1.The approximate Bayes estimate of the parameters is given by(27)ϑ^GE=Eϑϑ−ε|x−1/ε=1M−N∑l=N+1Mϑl−ε−1/ε.MCMC HPD credible interval algorithm is as follows:(1)
Sortσ∗,λ∗, and γ∗ in rising values.(2)
The lower bounds ofσ,λ, and γ in the rank M−N∗α/2.(3)
The lower bounds ofσ,λ, and γ in the rank M−N∗1−α/2.(4)
Iterate the previous stepsM times. Get the average value of the lower and upper bounds of σ,λ, and γ.


## 4.1. Maximum Likelihood Method

The likelihood function corresponding to equation (7) is given by(13)Lσ,λ,γ=n!σγλγn∏i=1nxiγ−1exp−∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,x∈ℝ.The log-likelihood function corresponding to equation (13) is given by(14)ℓ=logn!+nlogσ+nlogγ−nγlogλ+nγ−1∑i=1nlogxi−∑i=1nxiλγ+∑i=1nlog1−exp−xiλγ+∑i=1nlog2−1−σ1−exp−xiλγ−2∑i=1nlog1−1−σ1−exp−xiλγ.The equation in equation (14) can be solved by using Newton–Raphson method. The corresponding partial derivatives are(15)∂ℓ∂λ=γλn+1λ∑i=1nxixiλ−1+γ−∑i=1ne−xi/λγxixi/λ−1+γ1−e−xi/λγ+1−σ∑i=1ne−xi/λγxixi/λ−1+γ2−1−e−xi/λγ1−σ−2∑i=1ne−xi/λγxixi/λ−1+γ1−1−e−xi/λγ1−σ,∂ℓ∂γ=n1γ−logλ+∑i=1nlogxi−∑i=1nlogxiλxiλγ+∑i=1ne−xi/λγlogxi/λxi/λγ1−e−xi/λγ+1−σ2∑i=1ne−xi/λγlogxi/λxi/λγ1−1−e−xi/λγ1−σ−∑i=1ne−xi/λγlogxi/λxi/λγ2−1−e−xi/λγ1−σ,And,(16)∂ℓ∂σ=nσ−2∑i=1n1−e−xi/λγ1−1−e−xi/λγ1−σ+∑i=1n1−e−xi/λγ2−1−e−xi/λγ1−σ.The maximum likelihood estimators for the parameters of the Weibull distributionλ,γ, and σ are the solutions of the above nonlinear equations.Then, we can calculate the asymptotic confidence intervals of the parametersσ,λ, and γ. The covariance matrix of the observed variance for the MLEs of the parameters V^=σi,j,i,j=1,2,3 was assumed as follows:(17)V^=−∂2l∂σ2∂2l∂σ∂λ∂2l∂σ∂γ∂2l∂λ∂σ∂2l∂λ2∂2l∂λ∂γ∂2l∂γ∂σ∂2l∂γ∂λ∂2l∂γ2σ=σ^ML,λ=λ^ML,γ=γ^ML−1.1001−α% two-sided approximate confidence intervals for the parameters σ, λ, and γ are then given by(18)σ^±zα/2Vσ^,λ^±zα/2Vλ^,and(19)γ^±zα/2Vγ^,respectively, where Vσ^, Vλ^, and Vγ^ are the estimated variances of σ^ML, λ^ML, and γ^ML, which are given by the diagonal elements of V^, and zα/2 is the upper α/2 percentile of the standard normal distribution.Next, obtaining the bootstrap C.I. for boot-p for the unknown parametersϕ=σ,λ,γ, we apply the following algorithms (for more details, see [11, 12]).Boot-p interval’s algorithm is as follows:Step 1. Generate x1:n,x2:n,…,xn:n from the Weibull claim distribution and derive an estimate ϕ^ of ϕ.Step 2. Generate another sample x1:n∗,x2:n∗,…,xn:n∗ using ϕ^. Then, derive the updated bootstrap estimate ϕ^∗ from ϕ.Step 3. Repeat step 2 with a given number B of repetitions.Step 4. Using F^x=Pϕ^∗≤x, i.e., the CDF of ϕ^∗, the 1001−t% C.I. of ϕ is given by(20)ϕ^Boot−pα2,ϕ^Boot−p1−α2,whereϕ^Boot−px=F^−1x and x is prefixed.


## 4.2. Bayesian Estimation

Bayesian inference is an appropriate method to work with the full samples of the Weibull claim distribution. Given that Weibull claim distributions are so rare, prior information is very useful. We assume thatσ,λ, and γ are R.V.s following prior DFs: Uniform σ;0,aσ, Uniform λ;0,aλ, and Uniform γ;0,aγ, respectively. The posterior DF of σ,λ,andγ and the data under the uniform priors can take the form as follows:(21)πu∗σ,λ,γ|x∝Lσ,λ,γ|xπ1σ,λ,γ,=n!Juaσaλaγσγλγn∏i=1nxiγ−1exp−∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,where J1 is the normalizing constant. Next, we suppose that σ,λ, and γ are R.V.s that follow the prior DF Gamma σ;a1,b1, Gamma λ;a2,b2, and Gamma γ;a3,b3, respectively, where ai and bi are positive constants and i=1,2,3. The posterior DF of σ,λ,γ and the data under the Gamma priors can take the forms as follows:(22)πg∗σ,λ,γ|x∝Lσ,λ,γ|xπ1σ,λ,γ,=n!σa1−1λa2−1γa3−1Jgσγλγn∏i=1nxiγ−1×exp−b1σ+b2λ+b3γ+∑i=1nxiλγ×∏i=1n1−exp−xi/λγ2−1−σ1−exp−xi/λγ1−1−σ1−exp−xi/λγ2,where Jg is the marginal probability DF of x.


## 4.3. MCMC Method

In the following algorithm, we use Metropolis Hastings (M-H) procedure with normal DF to simulate samples from the distributions:(1)
Set the initial valuesσ0,λ0, and γ0. Then, simulate sample of size n from Weibull claim distribution, next let l=1.(2)
Simulateσ∗,λ∗, and γ∗. using the proposal distributions Nσl−1,Varσ^, Nλl−1,Varλ^, and Nγl−1,Varγ^.(3)
Obtain the probabilityr=minπ∗σ∗,λ∗,γ∗/π∗σl−1,λl−1,γl−1,1.(4)
SimulateU from Uniform (0, 1).(5)
IfU<r, then σl,λl,γl=σ∗,λ∗,γ∗. If U≥r, then σl,λl,γl=σl−1,λl−1,γl−1.(6)
Setl=l+1.(7)
Iterate Steps 2–6,M repetitions, and get σl,λl, and γl for l=1,…,M.In order to conduct a Bayesian analysis, usually quadratic loss function is considered. A very popular quadratic loss is the squared error loss function given byLSEϑ,ϑ^=ϑ−ϑ^2, where ϑ^ is an estimate of the unknown parameter ϑ against SE loss function. By using the generated random samples from the above Gibbs sampling technique and for N is the n burn, then the Bayes estimator of ϑ, say ϑ^SE, can be obtained as(23)ϑ^SE=Eϑϑ|x=1M−N∑l=N+1Mϑl.The second loss function is the LINEX loss function, given by(24)LLEϑ,ϑ^=expρϑ−ϑ^−ρϑ−ϑ^−1,ρ≠0.The approximate Bayes estimate ofϑ=σ,λ,γ under LE loss function based on the Gibbs sampling technique becomes(25)ϑ^LE=−1ρlogEϑexp−ρϑ|x=−1ρlog∑l=N+1Mexp−ρϑlM−N.Finally, the general entropy (GE) loss function is given by(26)LGEϑ,ϑ^=ϑ^ϑε−εlogϑ^ϑ−1.The approximate Bayes estimate of the parameters is given by(27)ϑ^GE=Eϑϑ−ε|x−1/ε=1M−N∑l=N+1Mϑl−ε−1/ε.MCMC HPD credible interval algorithm is as follows:(1)
Sortσ∗,λ∗, and γ∗ in rising values.(2)
The lower bounds ofσ,λ, and γ in the rank M−N∗α/2.(3)
The lower bounds ofσ,λ, and γ in the rank M−N∗1−α/2.(4)
Iterate the previous stepsM times. Get the average value of the lower and upper bounds of σ,λ, and γ.


## 5. Simulation Study

The simulation study is conducted as follows:(1)
Random samples of sizen=25,30,…,100 from the Weibull claim distribution are simulated.(2)
Parameters were estimated using maximum likelihood and Bayesian methods.(3)
Two LINEX loss functions are used:LE1 when ρ=−0.1 and LE2 when ρ=0.5.(4)
M iterations are performed to obtain the absolute biases and expected risk (ER) of these estimators.(5)
The point estimates of the parameters using MLE and MCMC methods are obtained.(6)
We obtain 95% and 90% HPD credibility intervals using MCMC methods.(7)
The biases and ERs are, respectively, given by(28)Biasϑ^=1M∑i=1Mϑ^i−ϑ.and(29)ERϑ^=1M∑i=1Mϑ^i−ϑ2.The MLE and Bayes simulation results of the Weibull claim distribution parametersλ,γ, and σ are presented in Tables 1 and 2.Table 1
Point estimation of the parametersλ=0.25,γ=2.5, and σ=1.77.
PointnPar.MLSELE1LE2GE25σ0.77310.49480.4990.47170.40040.51890.24070.24480.21760.14621.10620.17740.18250.14980.1106β1.18542.35152.35622.32522.3245−1.3166−0.1505−0.1459−0.1769−0.17764.54580.21170.21040.21940.2243γ1.22351.62911.63161.61531.6099−0.5508−0.1453−0.1428−0.159−0.16451.32220.05910.05870.06170.063550σ0.76680.34210.34440.32960.28420.51270.08790.09030.07550.03011.03570.08440.08650.07350.0598β1.21942.47332.47752.44982.4511−1.2826−0.0288−0.0246−0.0523−0.0514.92970.16420.16420.16430.1667γ1.16091.73141.73331.72081.7177−0.6135−0.043−0.041−0.0535−0.05671.00770.04290.04310.04230.0428200σ0.77880.23810.23870.2350.21850.5246−0.016−0.0154−0.0191−0.03561.01160.02350.02390.02160.0198β1.19822.57962.58132.57052.5715−1.30390.07760.07920.06840.06954.68890.08990.09040.08760.0884γ1.24231.77351.77391.77061.7699−0.5321−0.0009−0.0004−0.0038−0.00451.26190.01870.01880.01860.0186First line represents estimate, second line represents bias, and third line represents ER.Table 2
Interval estimation of the parametersλ=0.25,γ=2.5, and σ=1.77.
Par.IntervalMLBootHPDSHPDLE1HPDLE2HPDGE25σ0.49041.05570.0324.4190.0441.1850.04391.19640.04341.11680.03081.03890.56544.3871.1411.15251.07341.00810.53511.0110.062.3930.11.0990.09991.11640.0981.06850.0770.96560.4762.3330.9991.01650.97050.8886β0.90391.46690.3217.4271.2863.0361.29123.04241.26113.01421.2463.01780.5637.1061.751.75111.75311.77170.94841.42240.3395.8611.5622.9661.56912.96871.51982.95121.49562.95440.4745.5221.4041.39971.43131.4588γ0.94311.5040.6324.6981.2251.9211.22621.92411.21891.90261.2141.8990.5614.0660.6960.69790.68370.6850.98741.45970.6722.9481.2941.9161.29551.91891.28441.89041.2771.88380.47232.2760.6220.62340.6060.606850σ0.4761.05750.0273.9650.0551.1340.0551.14450.05481.07080.04730.95570.58153.9381.0791.08941.0160.90840.5221.01160.062.6210.0941.0160.09411.02930.09360.93730.07330.78550.48952.5610.9220.93520.84370.7122β0.86531.57360.3277.5971.5853.2931.58623.29731.58083.26361.57883.27190.70837.271.7081.7111.68281.69310.92131.51760.3486.8791.7693.2161.77063.22081.75263.18541.74493.19360.59636.5311.4471.45021.43271.4487γ0.83161.49020.6393.4691.3312.1241.33182.12941.32762.0981.32512.09630.65852.830.7930.79760.77040.77120.88371.43810.6742.241.3672.0781.36832.08051.35872.0611.35672.0590.55441.5660.7110.71230.70220.7023200σ0.4941.06350.0313.9460.0780.5380.07810.54020.07790.52440.07010.46690.56953.9150.460.46210.44640.39680.5391.01850.0562.5610.0960.4040.09650.40560.09580.39580.0830.36980.47952.5050.3080.30910.30.2868β0.92251.47390.3247.6271.9293.1291.93093.13241.91883.11031.91713.11520.55157.3031.21.20151.19161.19810.96611.43030.3546.492.1382.9832.1382.98682.12622.96042.12572.96530.46426.1360.8450.84880.83410.8395γ0.96461.52010.6334.6041.4992.0371.49912.03821.49712.03231.49622.03160.55553.9710.5380.5390.53520.53541.00851.47620.672.8831.5481.9671.54861.96771.54671.96241.5461.96160.46772.2130.4190.41910.41570.4157The first, second, third, and fourth lines show a 95% credible HPD interval, the corresponding 95% width, 90% credible HPD interval, and the corresponding 95% width of the parameter, respectively.


### 5.1. Application of the Weibull Claim Model

The data set represents the lifetimes of Kevlar 49/epoxy strands subjected to constant sustained pressure at 90% stress level until strand failure. For previous studies on data, see [13–16]. The data are as follows: 0.0251, 0.0886, 0.0891, 0.2501, 0.3113, 0.3451, 0.4763, 0.5650, 0.5671, 0.6566, 0.6748, 0.6751, 0.6753, 0.7696, 0.8375, 0.8391, 0.8425, 0.8645, 0.8851, 0.9113, 0.9120, 0.9836, 1.0483, 1.0596, 1.0773, 1.1733, 1.2570, 1.2766, 1.2985, 1.3211, 1.3503, 1.3551, 1.4595, 1.4880, 1.5728, 1.5733, 1.7083, 1.7263, 1.7460, 1.7630, 1.7746, 1.8275, 1.8375, 1.8503, 1.8808, 1.8878, 1.8881, 1.9316, 1.9558, 2.0048, 2.0408, 2.0903, 2.1093, 2.1330, 2.2100, 2.2460, 2.2878, 2.3203, 2.3470, 2.3513, 2.4951, 2.5260, 2.9911, 3.0256, 3.2678, 3.4045, 3.4846, 3.7433, 3.7455, 3.9143, 4.8073, 5.4005, 5.4435, 5.5295, 6.5541, and 9.0960.Table3 presents the point estimation of the Kevlar 49/epoxy strands parameters while Table 4 presents the corresponding interval estimation. Table 5 compares the Weibull claim distribution based on some detection criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and consistent Akaike information criterion (CAIC). The goodness-of-fit results of the Weibull claim model (W-claim) are compared with those of some other models, including the Exponential claim distribution (Exp-claim), the Weibull distribution (W-D), and the exponential distribution (EXP-D). Table 6 compares the W-claim with the Kolmogorov–Smirnov test for one sample.Table 3
Point estimation of the Kevlar 49/epoxy strands parameters.
PointNPar.MLSELE1LE2GE76σ0.35410.43190.43190.43190.4318−0.01070.07780.07780.07780.07760.00760.00610.00610.00610.0061β5.80495.76855.76865.76815.76840.3423−0.0364−0.0363−0.0368−0.03650.08550.0020.0020.00210.002γ1.06190.98620.98630.98620.98620.0597−0.0757−0.0757−0.0757−0.07580.00820.00580.00580.00580.0058First line represents estimate, second line represents bias, and third line represents ER.Table 4
Interval estimation of the Kevlar 49/epoxy strands parameters.
Par.IntervalMLBootHPDSHPDLE1HPDLE2HPDGE76σ0.18310.52510.2130.50.4180.4420.41760.44150.41760.44150.41740.44150.3420.2870.0240.02390.0240.02410.21020.49810.2220.4910.4220.4410.42170.44120.42170.44120.42140.44120.28790.2690.0190.01950.01960.0198β5.23176.37815.3376.2845.7195.8245.71865.82415.71865.82365.71865.82391.14640.9470.1050.10550.1050.10535.32246.28745.3636.2635.7295.8125.72875.81195.72875.81125.72875.81170.96510.90.0830.08320.08250.083γ0.88391.23990.9091.2120.9731.0090.97321.00860.97321.00830.97321.00810.3560.3030.0360.03530.03510.03490.91211.21180.9161.2040.9751.0020.97531.00230.97531.00230.97531.00230.29970.2880.0270.0270.0270.027The first, second, third, and fourth lines show a 95% credible HPD interval, the corresponding 95% width, 90% credible HPD interval, and the corresponding 95% width of the parameter, respectively.Table 5
Relative quality of the W-claim vs. competing models.
ModelAICCAICBICHQICW-claim (0.0087, 0.4494, 0.0497)246.5292246.8625253.5214249.3236Exp-claim (1.1915, 1.3556)249.3394249.5038254.0009251.2024W-D (1.3348, 2.1229)249.0626249.227253.7241250.9255EXP-D (1.9587)256.2287256.2827258.5594257.1601Table 6
One-sample Kolmogorov–Smirnov test.
ModelKSp valueW-claim (0.0087, 0.4494, 0.0497)0.0789990.700Exp-claim (1.1915, 1.3556)0.0914960.518W-D (1.3348, 2.1229)0.107340.322EXP-D (1.9587)0.16640.026


## 5.1. Application of the Weibull Claim Model

The data set represents the lifetimes of Kevlar 49/epoxy strands subjected to constant sustained pressure at 90% stress level until strand failure. For previous studies on data, see [13–16]. The data are as follows: 0.0251, 0.0886, 0.0891, 0.2501, 0.3113, 0.3451, 0.4763, 0.5650, 0.5671, 0.6566, 0.6748, 0.6751, 0.6753, 0.7696, 0.8375, 0.8391, 0.8425, 0.8645, 0.8851, 0.9113, 0.9120, 0.9836, 1.0483, 1.0596, 1.0773, 1.1733, 1.2570, 1.2766, 1.2985, 1.3211, 1.3503, 1.3551, 1.4595, 1.4880, 1.5728, 1.5733, 1.7083, 1.7263, 1.7460, 1.7630, 1.7746, 1.8275, 1.8375, 1.8503, 1.8808, 1.8878, 1.8881, 1.9316, 1.9558, 2.0048, 2.0408, 2.0903, 2.1093, 2.1330, 2.2100, 2.2460, 2.2878, 2.3203, 2.3470, 2.3513, 2.4951, 2.5260, 2.9911, 3.0256, 3.2678, 3.4045, 3.4846, 3.7433, 3.7455, 3.9143, 4.8073, 5.4005, 5.4435, 5.5295, 6.5541, and 9.0960.Table3 presents the point estimation of the Kevlar 49/epoxy strands parameters while Table 4 presents the corresponding interval estimation. Table 5 compares the Weibull claim distribution based on some detection criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and consistent Akaike information criterion (CAIC). The goodness-of-fit results of the Weibull claim model (W-claim) are compared with those of some other models, including the Exponential claim distribution (Exp-claim), the Weibull distribution (W-D), and the exponential distribution (EXP-D). Table 6 compares the W-claim with the Kolmogorov–Smirnov test for one sample.Table 3
Point estimation of the Kevlar 49/epoxy strands parameters.
PointNPar.MLSELE1LE2GE76σ0.35410.43190.43190.43190.4318−0.01070.07780.07780.07780.07760.00760.00610.00610.00610.0061β5.80495.76855.76865.76815.76840.3423−0.0364−0.0363−0.0368−0.03650.08550.0020.0020.00210.002γ1.06190.98620.98630.98620.98620.0597−0.0757−0.0757−0.0757−0.07580.00820.00580.00580.00580.0058First line represents estimate, second line represents bias, and third line represents ER.Table 4
Interval estimation of the Kevlar 49/epoxy strands parameters.
Par.IntervalMLBootHPDSHPDLE1HPDLE2HPDGE76σ0.18310.52510.2130.50.4180.4420.41760.44150.41760.44150.41740.44150.3420.2870.0240.02390.0240.02410.21020.49810.2220.4910.4220.4410.42170.44120.42170.44120.42140.44120.28790.2690.0190.01950.01960.0198β5.23176.37815.3376.2845.7195.8245.71865.82415.71865.82365.71865.82391.14640.9470.1050.10550.1050.10535.32246.28745.3636.2635.7295.8125.72875.81195.72875.81125.72875.81170.96510.90.0830.08320.08250.083γ0.88391.23990.9091.2120.9731.0090.97321.00860.97321.00830.97321.00810.3560.3030.0360.03530.03510.03490.91211.21180.9161.2040.9751.0020.97531.00230.97531.00230.97531.00230.29970.2880.0270.0270.0270.027The first, second, third, and fourth lines show a 95% credible HPD interval, the corresponding 95% width, 90% credible HPD interval, and the corresponding 95% width of the parameter, respectively.Table 5
Relative quality of the W-claim vs. competing models.
ModelAICCAICBICHQICW-claim (0.0087, 0.4494, 0.0497)246.5292246.8625253.5214249.3236Exp-claim (1.1915, 1.3556)249.3394249.5038254.0009251.2024W-D (1.3348, 2.1229)249.0626249.227253.7241250.9255EXP-D (1.9587)256.2287256.2827258.5594257.1601Table 6
One-sample Kolmogorov–Smirnov test.
ModelKSp valueW-claim (0.0087, 0.4494, 0.0497)0.0789990.700Exp-claim (1.1915, 1.3556)0.0914960.518W-D (1.3348, 2.1229)0.107340.322EXP-D (1.9587)0.16640.026


## 6. Discussion and Future Framework

The experimental side showed, depending on the two statistical criteria, the mean square errors and the bias in estimating the parameters, that the Bayes method ranks first in computing the parameter estimates and that the MLE method ranks second, especially for small samples. As the sample size increases, the MSE values for each experiment decrease. The values of the cumulative distribution are between zero and one, and they increase and are directly proportional to time. Based on the test results, it was shown that the proposed distribution best represents the data compared with competing distributions based on some AIC, BIC, HQIC, and CAIC. The goodness-of-fit results of the Weibull claim model are compared with Exp-claim, W-D, and EXP-D. The results in Tables5 and 6 suggest that the W-claim distribution provides a better fit than other competing models and could be chosen as a suitable model for analyzing heavy-tail engineering data.We recommend the use of the Bayes method for parameter estimation. In the future, it is possible to expand the scientific aspects associated with the application, such as the medical, engineering, and industrial aspects. Other estimation methods can also be used to estimate the parameters of the proposed new Weibull claim model and compare them with the methods used in this study.


## 7. Concluding Remarks

In this paper, we introduced a new extension of the Weibull distribution, the Weibull claim model. We determined the maximum likelihood estimators for the parameters of the Weibull claim distribution and performed a Bayesian Monte Carlo simulation study. The performances of the Bayesian estimators are better than those of the corresponding ML estimators. The new Weibull claim model outperformed many other old models. The performance of the Bayesian estimates is more excellent than that of the corresponding ML estimators. We found that the proposed model provides a better characterization of industry events by analyzing the Kevlar 49/epoxy strands data. Therefore, it can be a perfect model for predicting future cases.



---

*Source: 8729529-2022-05-04.xml*