# Robust Assessing the Lifetime Performance of Products with Inverse Gaussian Distribution in Bayesian and Classical Setup

Authors: Abdullah Ali H. Ahmadini; Amara Javed; Sohail Akhtar; Christophe Chesneau; Farrukh Jamal; Shokrya S. Alshqaq; Mohammed Elgarhy; Sanaa Al-Marzouki; M. H. Tahir; Waleed Almutiry
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The inverse Gaussian (Wald) distribution belongs to the two-parameter family of continuous distributions having a range from 0 to ∞ and is considered as a potential candidate to model diffusion processes and lifetime datasets. Bayesian analysis is a modern inferential technique in which we estimate the parameters of the posterior distribution obtained by formally combining a prior distribution with an observed data distribution. In this article, we have attempted to perform the Bayesian and classical analyses of the Wald distribution and compare the results. Jeffreys' and uniform priors are used as noninformative priors, while the exponential distribution is used as an informative prior. The analysis comprises finding joint posterior distributions, the posterior means, predictive distributions, and credible intervals. To illustrate the entire estimation procedure, we have used real and simulated datasets, and the results thus obtained are discussed and compared. We have used the Bayesian specialized Open BUGS software to perform Markov Chain Monte Carlo (MCMC) simulations using a real dataset.


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## Body

## 1. Introduction

In probability theory, the inverse Gaussian distribution (IGD), also known as the Wald distribution, belongs to the two-parameter family of continuous distributions with support 0 to ∞ [1]. The concept of Brownian motion is applicable in describing the inherent process of many phenomena, particularly in the natural and physical sciences. The time in which a Brownian motion with a positive drift reaches a fixed value is distributed as an IGD.The probability density function, or density for short, of the IGD is given by(1)fx;μ,λ=λ2πx31/2exp−λx−μ22μ2x,x>0,μ>0,λ>0.Here,μ is the location and λ is the shape parameter. The IGD approaches the normal distribution as λ⟶∞.A brief retrospective on the IGD is now proposed. To begin, the author in [2] studied the inverse Gaussian as a model to study Brownian motion. The author in first considered its basic features of statistical properties and found certain similarities in its statistical analysis. The authors in [3] proposed it as a lifetime model to be applied in situations where the initial failure rate was high. The author in [4] considered estimating the inverse Gaussian μ,λ model in a Bayesian framework. He also discussed that the estimation becomes very difficult when there is no natural conjugate prior. The authors in [5] considered the parameterization when ψ=1/μ and λ and evaluated Bayes estimates using uninformative reference and natural conjugate priors. The authors in [6] indicated that in [5] approach, the posterior mean of 1/ψ does not exist, so the Bayesian estimate of the mean of the distribution is not available. The author in [7] performed the estimation of the reliability function from the work done on IG μ,λ parameterization. The authors in [8] evaluated estimates for λ assuming Jeffreys’ prior and its posterior density by using the Gibbs sampling technique when μ is known.Because of its shapes, the related density may also be considered as a good competitor to the Gamma, Weibull, and log-normal distributions. Various sampling theory inferences with the IGD are studied by [9–12] among others. The authors in [13] treated some applications in marketing, while applications of IG in life testing are considered by [2, 3].The authors in [14] investigated Bayesian estimation for the parameters of the IGD distribution. They emphasized the MCMC technique and gave a complete implementation of the Gibbs sampler algorithm. The author in [5] obtained some Bayesian results for the inverse Gaussian family of distributions with a noninformative reference prior as well as the natural conjugate prior. The authors in [6] derived Bayesian results for the IGD by using a proper prior under reparameterization with reference to the distribution mean and of the inverse of the squared variation coefficient, for obtaining Bayes estimates as well as of their inverses. The author in [9] presented a report on some statistical properties of the IGD distribution when the parameters are confined to 0,∞. A good review of the advantages of using Bayesian methods may be found in [15–23]. The posterior distributions often have complex multidimensional forms that require using Markov Chain Monte Carlo (MCMC) methods to get results [16, 24–27]. In recent years, the use of Markov Chain Monte Carlo (MCMC) methods has gained much popularity [28–30]. Most recently, the author in [31] has considered in detail the q-Weibull distribution for classical and Bayesian analyses, which also serves as a motivation to conduct this study. Keeping in view the extensive literature on the importance of Bayesian analysis and the importance of the Wald distribution, we have attempted to present the Bayesian analysis of the Wald distribution. Jeffreys’ and uniform priors are used as noninformative priors, while the exponential distribution is used as an informative prior. The analysis comprises finding joint posterior distributions, the posterior means, predictive distributions, and credible intervals. To illustrate the entire estimation procedure, we have used real and simulated datasets, and the results thus obtained are discussed and compared.The literature reveals that a lot of authors have studied classical distributions, including the Wald distribution, in a classical framework. We have observed a variety of applications of the Wald distribution and the capability of the Bayes methods to incorporate the prior information of the model parameters. To the best of our knowledge and belief, the Wald distribution has not yet been studied in a Bayesian framework despite its potential applications. Therefore, to cover this gap present in the literature, we have attempted to perform the Bayesian analysis of the Wald distribution in this article.Here is the break-up of the study. Section2 considers the frequentist analysis of the inverse Gaussian distribution using the MLE method and computes the standard errors associated with the classical estimates. A numerical example is presented in Section 3. Section 4 carries out the Bayesian analysis of the IGD assuming the uniform, Jeffreys’, and subjective informative priors. It is supposed that the parameters of the IGD follow exponential distribution(s). The convergence diagnostic is given in Section 5. The predictive inference of the inverse Gaussian distribution is presented in Section 6. Comparison between the frequentist and Bayesian approaches is performed in Section 7. The simulation study is performed in Section 8 to justify the results.


## 2. Maximum Likelihood (ML) Estimation

Letx1,x2,…,xn constitute a random sample of size n from the IGD. The likelihood function is(2)px|μ,λ=λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi.The logarithmic form of the likelihood function is(3)l=lnpx|μ,λ=n2lnλ−n2ln2π−12∑i=1nlnxi3−λ2μ2∑i=1nxi−μ2xi.We differentiate this expression w.r.t the unknown parametersμ and λ, and equating the resulting equations to zero to maximize l, we get(4)μ^=∑i=1nxin=X¯,and,λ^=n∑i=1nxi−1−nx¯−1.They are the required ML estimates of the parameterμ and λ of the IGD.


### 2.1. Standard Errors of the ML Estimates

The main diagonal elements of the inverted Fisher information matrix (FIM) designate the variances of the ML estimates. Hence, we can find the standard errors of the ML estimates by calculating the square roots of the diagonal elements as(5)sμ^,λ^=1Iμ^,λ^.We find the elements of the Hessian matrix as follows. The second derivative ofl with respect to λ, μ, and μ,λ is given as(6)∂2l∂λ2=−n2λ2,∂2l∂μ2=−nλμ2and∂l∂μ∂λ=−∑i=1nμ−xiμ3=0.Fisher’s information matrix (F.I.M)Iμ,λ may be defined as follows: (7)Iμ,λ=−E∂2l∂μ2∂2l∂μ∂λ∂2l∂μ∂λ∂2l∂λ2,where(8)Iλ=−E∂2l∂λ2=n2λ2,Iμ=−E∂2l∂μ2=nλμ2andIμ,λ=−E∂2l∂μ∂λ=0.Hence,(9)Iμ,λ=−E∂2l∂μ2−E∂2l∂μ∂λ−E∂2l∂μ∂λ−E∂2l∂λ2=nλμ200n2λ2.The variance covariance matrixsμ^,λ^2 is(10)sμ^,λ^2=Iμ,λ−1=nλμ200n2λ2−1=1n2/2μ2λn2λ200nλμ2=μ2nλ2002λ2n.The uncorrelated parameters are displayed in the above matrix. The results are summarized in Table1.Table 1
Maximum likelihood estimators of the parameters with variances.
ParameterMaximum likelihood estimatorVariance of maximum likelihood estimatorμ∑i=1nxin=X¯X¯2∑i=1nxi−1−nx¯−12λn∑i=1nxi−1−nx¯−12∑i=1nxi−1−nx¯−1−2


## 2.1. Standard Errors of the ML Estimates

The main diagonal elements of the inverted Fisher information matrix (FIM) designate the variances of the ML estimates. Hence, we can find the standard errors of the ML estimates by calculating the square roots of the diagonal elements as(5)sμ^,λ^=1Iμ^,λ^.We find the elements of the Hessian matrix as follows. The second derivative ofl with respect to λ, μ, and μ,λ is given as(6)∂2l∂λ2=−n2λ2,∂2l∂μ2=−nλμ2and∂l∂μ∂λ=−∑i=1nμ−xiμ3=0.Fisher’s information matrix (F.I.M)Iμ,λ may be defined as follows: (7)Iμ,λ=−E∂2l∂μ2∂2l∂μ∂λ∂2l∂μ∂λ∂2l∂λ2,where(8)Iλ=−E∂2l∂λ2=n2λ2,Iμ=−E∂2l∂μ2=nλμ2andIμ,λ=−E∂2l∂μ∂λ=0.Hence,(9)Iμ,λ=−E∂2l∂μ2−E∂2l∂μ∂λ−E∂2l∂μ∂λ−E∂2l∂λ2=nλμ200n2λ2.The variance covariance matrixsμ^,λ^2 is(10)sμ^,λ^2=Iμ,λ−1=nλμ200n2λ2−1=1n2/2μ2λn2λ200nλμ2=μ2nλ2002λ2n.The uncorrelated parameters are displayed in the above matrix. The results are summarized in Table1.Table 1
Maximum likelihood estimators of the parameters with variances.
ParameterMaximum likelihood estimatorVariance of maximum likelihood estimatorμ∑i=1nxin=X¯X¯2∑i=1nxi−1−nx¯−12λn∑i=1nxi−1−nx¯−12∑i=1nxi−1−nx¯−1−2


## 3. Numerical Example

A real dataset is considered in this section that is analyzed by [14]. The dataset given in Table 2 denotes the active repair times (in hours) for an airborne communication transceiver.Table 2
Repair times (in hours) of 46 transceivers.
1.51.51.52.02.02.22.52.73.03.07.58.89.010.322.024.53.33.34.04.00.70.80.81.01.01.01.01.11.31.50.20.30.50.50.50.50.60.60.70.74.54.75.05.45.47.0————Using the data given above, the ML estimates along with their variance are computed and are given in Table3.Table 3
ML estimates and variances for the repair-time data.
ParametersMeanVarianceλ1.658850.01692μ3.606520.00260Here, we observe that the estimates are stable with very small standard errors.


## 4. Bayesian Analysis

### 4.1. Uninformative Bayesian Analysis Using the Uniform Prior

The uninformative uniform prior for the both parametersμandλ is defined as(11)pμ,λ∝1,μ>0,λ>0.The posterior distribution is given as(12)pμ,λ|x∝pμ,λ.px|μ,λ.Asμ and λ are considered independent, therefore, their joint prior distribution will be the product of their individual priors and may be defined as(13)pμ,λ|x∝pμ.pλ.px|μ,λ,⇒pμ,λ|x∝1.1.λ2πn/2∏i=1n∏1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0.


### 4.2. Uninformative Bayesian Analysis Using Jeffreys’ Prior

The positive square root of the determinant of the FIM is known as Jeffreys’ prior.(14)pJμ,λ=detFisher information matrix=Iμ,λ.Here,pJμ,λ denotes for Jeffreys’ prior. We carry on for Jeffreys’ prior as follows:The determinant of the FIM of the IGD is(15)Iμ,λ=nλμ200n2λ2⇒Iμ,λ=n22λμ2.Jeffreys’ priorpJμ,λ for unknown parameters μ,λ is given as follows:(16)pJμ,λ=n22λμ2⇒pJμ,λ∝λ−1μ−2=λ−1/2μ−1.Hence, for the unknown parametersμ and λ, the joint posterior distribution is given by(17)pJμ,λ|x∝pJμ,λ.px|μ,λ,pJμ,λ|x∝1λμ2.λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi,pJμ,λ|x=K−1μ−1λ1/2−1+n/2e−λ/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0,where K is denoted as the normalizing constant. It may be defined as(18)K=∫0∞∫0∞μ−1λ1/2−1+n/2e−λ/2μ2∑i=1nxi−μ2/xidμdλ.


### 4.3. Informative Bayesian Analysis Assuming the Exponential Prior

It is known that expert opinion can be incorporated into the analysis using the informative prior of an unknown parameter about a state of nature. To achieve this, we suppose an exponential prior for both parameters of the IGD.


#### 4.3.1. The Exponential Prior

Here, we assume that the prior distributions of both the parameters follow exponential distributions and are defined as(19)pλ=λe−h1λ,λ>0,h1>0,pμ=μe−h2μ,μ>0,h2>0.Here,h1 and h2 are known as the hyperparameters.So, the joint posterior distribution of the unknown parameterμ and λ is(20)pμ,λ|x∝pμ,λ.px|μ,λ.Asμ and λ are supposed to be independent,(21)pμ,λ=pμ.pλ.So, the joint posterior distribution of the parametersμ and λ is(22)pμ,λ|x∝pμ.pλ.px|μ,λ,⇒pμ,λ|x∝λe−h1λ.μe−h2μ.λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi,⇒pμ,λ|x∝μe−h2μ.λn/2+2−1.e−λh1+1/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0.Now, we observe that the two parameters of the IGD have independent distributions. Therefore, to make inferences, we will derive their marginal distributions regarding them. The marginal posterior distribution ofμ may be defined as(23)pμ|x=∫0∞pμ,λ|xdλ,0≤μ≤∞.Similarly, the marginal posterior distribution ofλ may be derived as follows:(24)pλ|x=∫0∞pμ,λ|xdμ,0≤λ≤∞.Some loss function is required to find the Bayesian estimates. So, we have considered the squared error loss function (SELF), which describes the posterior means as the Bayes estimates. That is,(25)μ^=Eμ|x=∫0∞μ.pμ|xdμ,λ^=Eλ|x=∫0∞λ.pλ|xdλ.Such expressions generally comprise complex structures, so the numerical methods are required to solve them. Therefore, to simulate data from the posterior distribution(s), we have used the MCMC technique by using OpenBUGS, and the resulting Bayes estimates, posterior risks, and 95% highest posterior density regions are given in Table4.Table 4
Bayes estimates ofμ,λ under uniform, Jefferys’, and exponential priors.
MeasuresUniform priorJeffreys’ priorExponential priorμλμλμλBayes estimates3.606521.73243.606521.73243.608341.65762Posterior risk0.012540.12510.012540.12510.012540.125195% HDR(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)The results show that the posterior estimates produced by the uniform, Jeffreys’, and exponential priors coincide a lot and have small standard errors. The posterior marginal densities of the model parameters are presented in Figure1.Figure 1
The posterior densities of the model parameters.The posterior marginal densities for the parameters of the model show a slight positive skewness for both of the parameters of the model.


## 4.1. Uninformative Bayesian Analysis Using the Uniform Prior

The uninformative uniform prior for the both parametersμandλ is defined as(11)pμ,λ∝1,μ>0,λ>0.The posterior distribution is given as(12)pμ,λ|x∝pμ,λ.px|μ,λ.Asμ and λ are considered independent, therefore, their joint prior distribution will be the product of their individual priors and may be defined as(13)pμ,λ|x∝pμ.pλ.px|μ,λ,⇒pμ,λ|x∝1.1.λ2πn/2∏i=1n∏1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0.


## 4.2. Uninformative Bayesian Analysis Using Jeffreys’ Prior

The positive square root of the determinant of the FIM is known as Jeffreys’ prior.(14)pJμ,λ=detFisher information matrix=Iμ,λ.Here,pJμ,λ denotes for Jeffreys’ prior. We carry on for Jeffreys’ prior as follows:The determinant of the FIM of the IGD is(15)Iμ,λ=nλμ200n2λ2⇒Iμ,λ=n22λμ2.Jeffreys’ priorpJμ,λ for unknown parameters μ,λ is given as follows:(16)pJμ,λ=n22λμ2⇒pJμ,λ∝λ−1μ−2=λ−1/2μ−1.Hence, for the unknown parametersμ and λ, the joint posterior distribution is given by(17)pJμ,λ|x∝pJμ,λ.px|μ,λ,pJμ,λ|x∝1λμ2.λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi,pJμ,λ|x=K−1μ−1λ1/2−1+n/2e−λ/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0,where K is denoted as the normalizing constant. It may be defined as(18)K=∫0∞∫0∞μ−1λ1/2−1+n/2e−λ/2μ2∑i=1nxi−μ2/xidμdλ.


## 4.3. Informative Bayesian Analysis Assuming the Exponential Prior

It is known that expert opinion can be incorporated into the analysis using the informative prior of an unknown parameter about a state of nature. To achieve this, we suppose an exponential prior for both parameters of the IGD.


### 4.3.1. The Exponential Prior

Here, we assume that the prior distributions of both the parameters follow exponential distributions and are defined as(19)pλ=λe−h1λ,λ>0,h1>0,pμ=μe−h2μ,μ>0,h2>0.Here,h1 and h2 are known as the hyperparameters.So, the joint posterior distribution of the unknown parameterμ and λ is(20)pμ,λ|x∝pμ,λ.px|μ,λ.Asμ and λ are supposed to be independent,(21)pμ,λ=pμ.pλ.So, the joint posterior distribution of the parametersμ and λ is(22)pμ,λ|x∝pμ.pλ.px|μ,λ,⇒pμ,λ|x∝λe−h1λ.μe−h2μ.λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi,⇒pμ,λ|x∝μe−h2μ.λn/2+2−1.e−λh1+1/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0.Now, we observe that the two parameters of the IGD have independent distributions. Therefore, to make inferences, we will derive their marginal distributions regarding them. The marginal posterior distribution ofμ may be defined as(23)pμ|x=∫0∞pμ,λ|xdλ,0≤μ≤∞.Similarly, the marginal posterior distribution ofλ may be derived as follows:(24)pλ|x=∫0∞pμ,λ|xdμ,0≤λ≤∞.Some loss function is required to find the Bayesian estimates. So, we have considered the squared error loss function (SELF), which describes the posterior means as the Bayes estimates. That is,(25)μ^=Eμ|x=∫0∞μ.pμ|xdμ,λ^=Eλ|x=∫0∞λ.pλ|xdλ.Such expressions generally comprise complex structures, so the numerical methods are required to solve them. Therefore, to simulate data from the posterior distribution(s), we have used the MCMC technique by using OpenBUGS, and the resulting Bayes estimates, posterior risks, and 95% highest posterior density regions are given in Table4.Table 4
Bayes estimates ofμ,λ under uniform, Jefferys’, and exponential priors.
MeasuresUniform priorJeffreys’ priorExponential priorμλμλμλBayes estimates3.606521.73243.606521.73243.608341.65762Posterior risk0.012540.12510.012540.12510.012540.125195% HDR(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)The results show that the posterior estimates produced by the uniform, Jeffreys’, and exponential priors coincide a lot and have small standard errors. The posterior marginal densities of the model parameters are presented in Figure1.Figure 1
The posterior densities of the model parameters.The posterior marginal densities for the parameters of the model show a slight positive skewness for both of the parameters of the model.


## 4.3.1. The Exponential Prior

Here, we assume that the prior distributions of both the parameters follow exponential distributions and are defined as(19)pλ=λe−h1λ,λ>0,h1>0,pμ=μe−h2μ,μ>0,h2>0.Here,h1 and h2 are known as the hyperparameters.So, the joint posterior distribution of the unknown parameterμ and λ is(20)pμ,λ|x∝pμ,λ.px|μ,λ.Asμ and λ are supposed to be independent,(21)pμ,λ=pμ.pλ.So, the joint posterior distribution of the parametersμ and λ is(22)pμ,λ|x∝pμ.pλ.px|μ,λ,⇒pμ,λ|x∝λe−h1λ.μe−h2μ.λ2πn/2∏i=1n1xi31/2e−λ/2μ2∑i=1nxi−μ2/xi,⇒pμ,λ|x∝μe−h2μ.λn/2+2−1.e−λh1+1/2μ2∑i=1nxi−μ2/xi;μ>0,λ>0.Now, we observe that the two parameters of the IGD have independent distributions. Therefore, to make inferences, we will derive their marginal distributions regarding them. The marginal posterior distribution ofμ may be defined as(23)pμ|x=∫0∞pμ,λ|xdλ,0≤μ≤∞.Similarly, the marginal posterior distribution ofλ may be derived as follows:(24)pλ|x=∫0∞pμ,λ|xdμ,0≤λ≤∞.Some loss function is required to find the Bayesian estimates. So, we have considered the squared error loss function (SELF), which describes the posterior means as the Bayes estimates. That is,(25)μ^=Eμ|x=∫0∞μ.pμ|xdμ,λ^=Eλ|x=∫0∞λ.pλ|xdλ.Such expressions generally comprise complex structures, so the numerical methods are required to solve them. Therefore, to simulate data from the posterior distribution(s), we have used the MCMC technique by using OpenBUGS, and the resulting Bayes estimates, posterior risks, and 95% highest posterior density regions are given in Table4.Table 4
Bayes estimates ofμ,λ under uniform, Jefferys’, and exponential priors.
MeasuresUniform priorJeffreys’ priorExponential priorμλμλμλBayes estimates3.606521.73243.606521.73243.608341.65762Posterior risk0.012540.12510.012540.12510.012540.125195% HDR(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)(2.7548, 5.5698)(1.4822, 2.1077)The results show that the posterior estimates produced by the uniform, Jeffreys’, and exponential priors coincide a lot and have small standard errors. The posterior marginal densities of the model parameters are presented in Figure1.Figure 1
The posterior densities of the model parameters.The posterior marginal densities for the parameters of the model show a slight positive skewness for both of the parameters of the model.


## 5. Convergence Diagnostics

The dynamic traces and time series plots are assessed to check the convergence. They are presented in Figure2. The simulated results confirm the convergence of the parameters of the underlying posterior distribution.Figure 2
Dynamic traces and posterior densities of the parameters.


## 6. Predictive Inference

We evaluate the predictive distributions to study the future behaviour of data. Using the posterior distributionpμ,λ|x based on the exponential distribution defined in Section 4.3 and the Wald distribution py|μ,λ as data model, the predictive distribution can be expressed as(26)py|x=∫0∞∫0∞py|μ,λ.pμ,λ|xdμdλ=∫0∞∫0∞λ2πy31/2e−λy−μ2/2μ2yμe−h2μλn/2+1e−λh1+1/2μ2∑i=1nxi−μ2/xidμdλ=12πy31/2∫0∞∫0∞λn+3/2μe−h2μe−λh1+1/2μ2∑i=1nxi−μ2/xi+y−μ2/2μ2ydμdλ=12πy31/2∫0∞∫0∞λn+3/2μe−h2μe−λh1+1/2μ2∑i=1nxi−μ2/xi+y−μ2/ydμdλ.Often, the predictive distributions do not follow the baseline distributions and do not have closed forms. Here, we observed that the predictive distribution is not in closed form. So, we will require numerical methods to evaluate multiple integrals to evaluate the above defined predictive distribution. This is accomplished by using the numerical procedures, and the resulting predicted and observed datasets are summarized in Table5 with graphical representation made in Figure 3.Table 5
Summary of the predicted and real datasets.
Types of dataMinimum1st quartileMedianMean3rd quartileMaximumObserved0.2000.8001.7503.6074.37524.5Predicted0.21370.76961.08112.25132.451016.1022Figure 3
Histograms and boxplots of the predicted and real datasets. (a) Boxplot of predicted data. (b) Boxplot of real data. (c) Histogram of real data. (d) Histogram of predicted data.
(a)(b)(c)(d)It shows that the predicted and observed datasets are much identical to each other.


## 7. Comparison of the Frequentist and Bayesian Approaches

To make a comparison of the Bayesian estimation method with the frequentist maximum likelihood estimation method, several model selection criteria, i.e., log-likelihood (LL), Akaike information criterion (AIC), and the Bayes information criterion (BIC) are used. That is a vital part of this study. The better the fit is, the lower the values of these criteria are. The ML estimates, noninformative Bayesian estimates, and informative Bayesian estimates with the values of the model selection criteria are evaluated and presented in Table6.Table 6
Summary of comparisons of estimation methods.
Estimation methodsPriorsλμLLAICBICML—1.6593.607−99.0477202.0953205.7526UMVUE—1.5513.607−99.0988202.1976205.8549Bayes Lindley—1.6234.118−99.2127202.4255206.0827Bayes methodsλ∼gamma3.3,2,μ∼gamma6.8,1.91.8593.354−99.2699202.5399206.1971λ∼exp0.6024,μ∼exp0.27781.6204.008−99.1577202.3155205.9728Jeffrey’s prior1.6584.210−99.2649202.5299206.1872Uniform prior1.6685.346−100.1742204.3484208.0057The results reveal that the values of the AIC and BIC computed by Bayesian estimates are the smallest as compared with those produced by frequentist estimates. It is noted that the Gamma prior has minimumAIC and BIC values for Bayes estimates as the expert opinion is also involved in it.


## 8. Simulation Study

Simulation studies help us to understand the type and behaviour of the underlying distributions. So, we generate data values for the parameters and estimate them based on the generated data. Here, the same parametric values have been used that are obtained by using the real dataset throughR codes, and the results are portrayed in Tables 7 and 8.Table 7
Estimates for 100000 simulations.
nμ^λ^Eμ^Eλ^SEμ^SEλ^LLμULμLLλULλ53.60651.65883.60204.14782.37389.77660.95659.88780.7425117.158103.60651.65883.60162.37241.67161.50681.38327.78390.86876.2250203.60651.65883.60401.94921.18890.71641.82856.43511.00783.7305503.606521.65883.60861.76570.75100.37292.35735.28911.18302.63341003.60651.65883.60611.71000.53020.24872.67564.74761.29122.2630Table 8
Estimates for 500000 simulations.
nμ^λ^Eμ^Eλ^SEμ^SEλ^LLμULμLLλULλ53.60651.65883.60334.14772.37409.32280.95589.85950.741617.2688103.60651.65883.60572.36721.68521.49421.38427.82310.87116.1343203.60651.65883.60441.95321.18720.71701.83486.42511.00903.7324503.60651.65883.60701.76500.75350.37242.35645.29091.18112.62931003.60651.65883.60751.71010.53110.24762.67764.75501.29262.2591The results show that the model will produce the same results if it continues to run on a similar pattern in the future. Moreover, the estimates become more stable if the numbers of simulations are increased.


## 9. Conclusion and Recommendations

The Bayesian inference for the parameters of the Wald distribution has been performed in this study and also compared the results with their classical counterparts. We have evaluated the maximum likelihood estimates for the comparison purpose. We used uniform and Jeffreys' priors as noninformative priors and the exponential distribution as informative prior. We have also discussed the predictive distribution as well. Simulation studies have also been conducted to verify the results. It has been witnessed that the results produced by using the Bayesian technique produce better results by yielding smaller AIC and BIC values. We have also witnessed close coordination between the observed and predicted datasets, which indicates that the Bayes methods are the potential replacements for their classical counterparts. The Bayesian methods are best suited for evaluating the lifetime data of any type of product. Future perspectives of the research may be to conduct such studies using other distributions that can model a variety of natural phenomena. We may also extend this study to include the generalized and multivariate distributions as well.



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*Source: 4582958-2021-10-04.xml*