## 3 Expectation Maximization Algorithm  
Estimating Weibull distribution parameters in the presence of censored data often poses challenges, as closed-form analytical solutions for the maximum likelihood equations are typically unavailable. Explicit formulae to directly solve parameter estimates are not feasible in this case. When dealing with censored data, the likelihood function becomes more intricate, incorporating both observed failure times and censored observations. The likelihood function includes terms representing the probabilities of observed failure times and the probabilities of failure times being censored. Due to the complexity of the problem, analytical solutions for the maximum likelihood equations are not viable. Instead, numerical methods and iterative algorithms are commonly employed to estimate the parameters, which maximize the likelihood function [21]. These methods iteratively update the parameter estimates until convergence is achieved, searching for the values that optimize the likelihood function. In this work, the expectation maximization (EM) algorithm was chosen among other numerical methods due to its demonstrated effectiveness in producing reliable results [22; 23]. The EM algorithm, introduced by [PERSON], [PERSON], and [PERSON] in 1977, is an iterative optimization algorithm [24]. It is widely used in reliability and survival analysis, as well as in other fields, such as machine learning, data mining, and bioinformatics [25; 26; 27]. The EM algorithm is a powerful statistical tool for estimating parameters in complex models, particularly in situations involving incomplete or censored data, where analytical solutions are challenging [28; 29]. The algorithm alternates between two steps: the E-step (expectation step) and the M-step (maximization step). In the E-step, the algorithm calculates the expected value of the log-likelihood function based on the current parameter estimates. In the M-step, it maximizes the expected value of the log-likelihood function concerning the parameters [24; 30]. The log-likelihood function for the complete data set \(X\) is denoted as \(l_{c}(x,\theta)\). When incomplete data are present, certain events are unknown, and the observed data set is represented as \(Y\), while \(Z\) represents the unknown data. Consequently, \(X\) can be expressed as a function of \((y,z)\). The EM algorithm with the presence of incomplete data can be summarized as follows:  
1. Initialization: The algorithm begins by initializing the model's parameters, denoted as \(\theta^{(0)}\). This initialization can be conducted randomly or with reasonable initial values. It is crucial to pay attention to the choice of initial values, as poor selection can result in slow algorithm convergence. Additionally, since the maximum likelihood equation can have multiple solutions corresponding to local maxima, the choice of initial values becomes significant. A comparative study on different strategies for choosing initial values was conducted by [PERSON] [31], highlighting the dependence of the strategy on the selection of initial solutions. 2. E-step (expectation step): The algorithm calculates the expected values of the missing or unobserved data in the E-step, given the current parameter estimates. This step involves computing the posterior probability distribution of the missing data, which represents the uncertainty about their values. The expectation is made concerning the conditional distribution of the missing data, conditioned on the observed data and the current parameter estimates. To perform the E-step, the algorithm typically utilizes the complete data likelihood function, which incorporates both the observed and missing data. However, since the missing data are unavailable, the algorithm computes the expectation of the complete data log likelihood instead. This expectation is often referred to as the \"Q-function\" (Equation (4)) [24]. \[Q\Big{(}\theta,\theta^{k}\Big{)}=E_{\theta^{k}}\left(l_{c}(x,\theta)\Big{|} y,\delta,\theta^{k-1}\right)\] (4)
3.