## 2 Reliability Analysis with Censored Data  
The Weibull distribution is widely utilized in reliability studies, survival analysis, and various other fields due to its versatility. It is commonly employed for modeling the failure rates of components and systems and estimating their lifetimes. The Weibull distribution can effectively fit data from diverse sources, including laboratory tests, field data, and warranty claims. Estimating its parameters can be accomplished through methods such as maximum likelihood estimation, enabling the development of models, which facilitate the predictions of future failures and reliability comparisons among different products or designs. Maximum likelihood estimation (MLE) is fundamental for estimating unknown parameters in statistical models [13; 14]. This approach involves determining the parameter values, which maximize the likelihood function, constructed based on the observed data and the parameters. The resulting estimates are frequently employed for making predictions and inferences about the underlying population from which the data were sampled. The likelihood function is established by considering the joint probability distribution of the observed data. In simpler terms, it quantifies the probability of obtaining the observed data for various parameter values [15]. The maximum likelihood estimation method seeks to identify the parameter values, which maximize this likelihood function, thereby providing the most plausible parameter estimates (Equation (1)). \[\text{max}L(\theta_{1},\theta_{2},\dots,\theta_{n})=max\prod_{i=1}^{n}f(\theta|x _{i}), \tag{1}\]  
where \(x_{i}=x_{1},x_{2},\dots,x_{n}\) is a sample of \(n\) independent observations of the random variable \(X\) from a distribution with probability density function \(f(x,\theta)\), and \(\theta_{i}=\theta_{1},\theta_{2},\dots,\theta_{n}\) is the vector of unknown parameters. In practical applications, maximizing the log-likelihood function is often more convenient than the likelihood directly [16; 17]. The log-likelihood function offers several advantages, as it is a monotonically increasing function of the likelihood and simplifies mathematical calculations. Consequently, the maximum likelihood estimation (MLE) problem can be transformed into a maximization problem of the log likelihood. Assuming that each failure time (\(t_{i}=t_{1},\,t_{2},\,\dots,\,t_{n}\)) represents an independent data point from the same representative population following the Weibull distribution with scale parameter \(\eta\) and shape parameter \(\beta\), the log-likelihood function for the Weibull distribution with complete data is expressed as follows (Equation (2)) [18]:  
\[\ln L(\eta,\beta)=\eta ln\beta-\eta\beta ln\eta+(\beta-1)\sum_{i=1}^{n}\left( lnt_{i}\right)-\sum_{i=1}^{n}\left(\frac{t_{i}}{\eta}\right)^{\beta} \tag{2}\]  
Once the log-likelihood function is defined, various optimization techniques, including numerical optimization algorithms, can be employed to identify the maximum function. The parameter values corresponding to this maximum are considered the maximum likelihood estimation (MLE) estimates. These estimates represent the most likely values for the parameters given in the observed data. By maximizing the log-likelihood function, we obtain parameter estimates, which provide the best fit to the data according to the Weibull distribution model. Historical failure data, documenting past failures, hold significant importance in reliability analysis, guiding the decision making related to maintenance strategies, equipment replacement or refurbishment, spare part stocking, and warranty policies. Through data analysis, optimal maintenance intervals can be determined, critical components that frequently fail can be identified, and the cost effectiveness of different maintenance approaches can be assessed. Historical fault data can be categorized into two types based on the availability of information: complete data and censored data [10; 19].