# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 4 Three Different Estimation Methods
### Bayesian Estimation  
The Bayesian technique is extremely helpful in the analysis of reliability since it allows for the incorporation of previous knowledge into the research process. This is important because the restricted availability of data is one of the major issues with reliability analysis. By apply the insightful before to the \(\alpha,\beta,\theta\), and \(\lambda\) parameters, which are prior gamma distributions. \(\alpha,\beta,\theta\), and \(\lambda\) have independent joint prior density functions that look like this:  
\[\Pi(\varphi)\propto\alpha^{q_{1}-1}\beta^{q_{2}-1}\lambda^{q_{3}-1}\theta^{q _{4}-1}e^{-(w_{1}a+w_{2}\beta+w_{3}\lambda+w_{4}\theta)};q_{j},\ w_{j}>0;\ j=1,\ldots,4. \tag{32}\]  
By determining suitable and superior values for the independent joint prior's hyper-parameters using the estimate and variance-covariance matrix of the MLE technique. By equating the gamma priors' mean and variance as the following equations, the estimated hyper-parameters can be determined as follows:  
\[q_{j}=\frac{\left[\frac{1}{L}\sum_{i=1}^{L}\hat{\varphi}_{i}^{j}\right]^{2}}{ \frac{1}{L-1}\sum_{i=1}^{L}\left[\hat{\varphi}_{i}^{j}-\frac{1}{L}\sum_{i=1}^{L }\hat{\varphi}_{i}^{j}\right]^{2}},\ j=1,\ldots,4, \tag{33}\]\[w_{j}=\frac{\frac{\mathbb{I}\sum_{i=1}^{L}\phi_{i}^{j}}{\frac{1}{L-1}\sum_{i=1}^{L }\left[\phi_{i}^{j}-\frac{1}{L}\sum_{i=1}^{L}\phi_{i}^{j}\right]^{2}},\,j=1, \ldots,4, \tag{34}\]  
where, \(L\) is the number of iteration and \(\phi^{1}=\hat{\alpha}\), \(\phi^{2}=\hat{\beta}\), \(\hat{\phi}^{4}=\hat{\theta}\) and \(\hat{\phi}^{3}=\hat{\lambda}\). The likelihood function and joint prior function combine to form the joint posterior density function of \(\varphi\). The joint posterior of the TLMW distribution is thus written as  
\[ \begin{split}\Pi(\varphi|x)\propto& e^{-2\sum_{i=1 }^{m}(\alpha x_{i}+\beta x_{i}^{\theta})}\prod_{l=1}^{n}\left(\alpha+\theta \beta x_{i}^{\theta-1}\right)\left[1-e^{-2(\alpha x_{i}+\beta x_{i}^{\theta}) }\right]^{\lambda-1}\\ &\alpha^{q_{1}-1}\beta^{q_{2}-1}\lambda^{n+q_{3}-1}\theta^{q_{4 }-1}e^{-(m_{1}\alpha+\alpha_{2}\beta+\alpha_{3}\lambda+\alpha_{4}\theta)}, \end{split}  \tag{35}\]  
utilising a function for squared error loss (SEL), which is the most typical function for symmetric loss. The Bayesian estimators of \(\hat{\varphi}\) based on the SEL function. \[ \begin{split} S(\hat{\varphi})&=E(\hat{\varphi}- \varphi)^{2}\\ &\int_{0}^{\infty}...\int_{0}^{\infty}(\hat{\varphi}-\varphi)^{2} \Pi(\varphi|x)d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}.\end{split}  \tag{36}\]  
The integrals provided by Equation (36) cannot be directly obtained, it should be emphasised. By employing the Markov chain Monte Carlo (MCMC) technique to estimate the value of integrals. Important sub-classes of MCMC approaches include Gibbs sampling and, within Gibbs samplers, a more generic Metropolis-Hesting. The Metropolis-Hasting (M-H) algorithm and Gibbs sampling are the two most often used MCMC applications. The MH technique, like acceptance-rejection sampling, presupposes that a candidate value can be generated from a proposal distribution for each iteration of the algorithm. The M-H algorithm claims that a candidate value from a normal proposal distribution can be generated for each iteration of the process, much like acceptance-rejection sampling claims. To obtain random samples of conditional posterior densities from the TLMW distribution, the M-H will be used inside Gibbs sampling.