## 2 Transformer Health Index Estimation Model
### Estimations Distribution Parameters Estimation  
There are several methods that can be used in order to estimate the Weibull and normal distributions parameters, such as the ordinary least square (OLS), weighted least square (WLS), maximum likelihood estimate (MLE), and method of moments (MOM) [33]. The OLS and the WLS methods are commonly used due to its simplicity [37]. The parameters estimation can be calculated by solving the simultaneous equations. The MLE and the MOM are the common methods that are used for engineering analyses, but both are computationally demanding [37]. Moreover, MLE and likelihood functions normally desire significant numbers of sample size, which develop unbiased minimum variance estimators as the sample size increases [38; 39]. In this paper, the MLE method was used for estimating the population parameters of a distribution. It is because MLE is an analytic maximization procedure, which is applicable to all form of data [34]. These methods also have approximate normal distributions and sample variances that can be utilized to produce confidence bounds likelihood functions to test the models and parameters' hypotheses. Suppose that \(x_{1}\), \(x_{2},\ldots,x_{n}\) are independent and identically distributed Weibull parameters, the random variables have probability density function \(f(x)\) expressed in Equation (1), where the parameters are assumed to be unknown. The MLE method was employed to estimate the parameters \(\alpha\) and \(\beta\). The likelihood function of \(x_{1}\), \(x_{2},\ldots,x_{n}\) can be constructed from Equation (2) and expressed in Equation (5) [40],  
\[L(\alpha,\beta)=\alpha^{n}\beta^{n}\left(\prod_{i=1}^{n}x_{i}\right)^{\beta-1} exp\left(-\alpha\sum_{i=1}^{n}x_{i}^{\beta}\right) \tag{5}\]  
Equation (6) can be obtained by taking the natural logarithmic transformation of Equation (5). \[lnL(\alpha,\beta)=nln\alpha+nln\beta+(\beta-1)\sum_{i=1}^{n}lnx_{i}-\alpha\sum _{i=1}^{n}x_{i}^{\beta} \tag{6}\]  
Differentiating Equation (6) with respect to \(\alpha\) and \(\beta\), respectively, yields Equations (7) and (8). \[\frac{\partial}{\partial\alpha}lnL(\alpha,\beta)=\frac{n}{\alpha}-\sum_{i=1}^ {n}x_{i}^{\beta} \tag{7}\]  
\[\frac{\partial}{\partial\alpha}lnL(\alpha,\beta)=\frac{n}{\beta}+\sum_{i=1}^ {n}lnx_{i}-\alpha\sum_{i=1}^{n}x_{i}^{\beta}lnx_{i} \tag{8}\]  
Hence, the MLE estimates \((\hat{\alpha},\hat{\beta})\) of \((\alpha,\beta)\) can be estimated from the following Equations (9) and (10). \[\frac{n}{\hat{\alpha}}=\sum_{i=1}^{n}x_{i}^{\beta} \tag{9}\]  
\[\frac{n}{\hat{\beta}}=\sum_{i=1}^{n}lnx_{i}=\hat{\alpha}\sum_{i=1}^{n}x_{i}^{ \hat{\beta}}lnx_{i} \tag{10}\]  
or equivalently  
\[\hat{\alpha}=\frac{n}{\sum_{i}^{n}x_{i}^{\hat{\beta}}} \tag{11}\]  
\[n+\hat{\beta}\sum_{i=1}^{n}lnx_{i}=\frac{n\hat{\beta}\sum_{i=1}^{n}x_{i}^{ \hat{\beta}}lnx_{i}}{\sum_{i=1}^{n}x_{i}^{\hat{\beta}}} \tag{12}\]  
Equations (11) and (12) can be numerically solved for \(\hat{\beta}\) and \(\hat{\alpha}\). Similarly, for normal distribution, the parameters \(\mu\) and \(\sigma\) were estimated using the MLE. The likelihood function of \(x_{1}\), \(x_{2},\ldots,x_{n}\) can be constructed from Equation (4) and it is expressed in Equation (13). \[L(\mu,\sigma)=\prod_{i=1}^{n}\frac{1}{\sigma\sqrt{2\pi}}exp\left(-\frac{1}{2 }\left[\frac{x_{i}-\mu}{\sigma}\right]^{2}\right) \tag{13}\]  
From Equation (13), the log-likelihood can be expressed based on (14). \[l(\mu,\sigma)=-nlog\sigma-\frac{n}{2}log2\pi-\frac{1}{2\sigma^{2}}\sum_{i=1}^ {n}(x_{i}-\mu)^{2} \tag{14}\]  
Differentiating Equation (14) with respect to \(\mu\) and \(\sigma\), respectively, yields Equations (15) and (16). \[lnL(\alpha,\beta)=nln\alpha+nln\beta+(\beta-1)\sum_{i=1}^{n}lnx_{i}-\alpha\sum _{i=1}^{n}x_{i}^{\beta} \tag{6}\]0  
\[lnL(\alpha,\beta)=nln\alpha+nln\beta+(\beta-1)\sum_{i=1}^{n}lnx_{i}-\alpha\sum _{i=1}^{n}x_{i}^{\beta} \tag{6}\]1Equating Equation (15) to zero, the MLE \((\hat{\mu}\,,\hat{\sigma})\) of \((\mu,\sigma)\) solutions are expressed in Equation (17).