## 3 Expectation Maximization Algorithm
The EM algorithm is a practical approach to addressing the challenge of right censored data in reliability analysis. The EM algorithm offers a robust framework for estimating the parameters of a chosen reliability model when working with censored data. In the case of right censored data, the algorithm employs an iterative process. It imputes the unobserved failure times based on the current parameter estimates and updates them using the imputed failure times. This iterative process enables the incorporation of censored data and enhances parameter estimation accuracy. During the E-step of the EM algorithm for right censored data in reliability studies, the survival probabilities are computed using the current parameter estimates for each observation. These survival probabilities represent the probability of survival beyond the censoring time for each observation. Subsequently, the survival probabilities are utilized to impute the failure times for the censored observations. The Q-equation, which characterizes the E-step in the EM algorithm for right censored data, can be expressed as follows, as presented in Ref. [18] (Equation (6)):  
Figure 1: Interactive process of the EM algorithm. \[ \begin{split} Q\Big{(}\theta,\theta^{k}\Big{)}=nln\beta-n\beta ln \eta+(\beta-1)\sum\limits_{i=1}^{n}\bigg{[}\delta_{i}lny_{i}+(1-\delta_{i}) \bigg{\{}lny_{i}+\frac{1}{\beta^{k}}exp\Big{(}\frac{y_{i}}{\eta^{k}}\Big{)}^{ \beta^{k}}\Gamma\bigg{[}0,\Big{(}\frac{y_{i}}{\eta^{k}}\Big{)}^{\beta^{k}} \bigg{]}\bigg{\}}\bigg{]}\\ -\frac{1}{\eta^{k}}\sum\limits_{i=1}^{n}\bigg{\{}\delta_{i}y_{i}^{ \beta}+(1-\delta_{i})\bigg{[}y_{i}^{\beta^{k}}+\Big{(}\eta^{k}\Big{)}^{\beta^{ k}}\bigg{]}\bigg{\}}\end{split} \] (6) where \[\Gamma(p,x)=\int\limits_{x}^{\infty}u^{p-1}e^{-u}du,\text{ is the incomplete gamma function.}\]  
This equation estimates the missing failure times based on the available survival probabilities. By imputing the failure times for the censored observations, the algorithm iteratively refines the parameter estimates, progressively improving the fit between the reliability model and the right censored data. As seen above, the M-step is designed to find the solution \(\theta^{k+1}\), which maximizes \(Q(\theta,\theta^{k+1})\). Once the failure times are imputed, the M-step of the EM algorithm proceeds to update the parameter estimates. In this step, the algorithm maximizes the expected complete data log likelihood, incorporating both observed and imputed failure times. By incorporating the imputed failure times, the algorithm effectively accounts for the censored information and provides more accurate parameter estimates. When applied to right censored data, the EM algorithm offers a robust approach for estimating reliability parameters in the presence of censored observations. It overcomes the limitations imposed by censoring, such as incomplete failure time information. It ensures that the analysis incorporates all available data to make reliable inferences about the reliability characteristics of the system or equipment under study. It is important to note that the specific implementation of the EM algorithm for censored data in reliability studies relies on the chosen reliability model, such as the Weibull distribution, and the assumptions made about the underlying distribution. Using appropriate statistical techniques can significantly enhance the analysis and estimation process. These procedures enable the modeling of the system based on the fit of a predefined distribution. In the following section, a case study will be presented to demonstrate the effectiveness of this approach in a real-world scenario.