## 2 Maximum Likelihood Estimation  
In the progressive Type-II censored model, a total of \(n\) units are included in a reliability test, with only \(m\) units being monitored until failure. Upon the occurrence of the first failure (\(x_{1}\)), a random selection of \(R_{1}\) units is removed from the remaining \(n-1\) units. This procedure is repeated for each subsequent failure, resulting in the removal of \(R_{2}\) units following the second failure (\(x_{2}\)), and so forth, until \(R_{m}\) units are removed after the \(m\)-th failure. The censoring scheme, denoted as \(R=(R_{1},\ldots,R_{m})\), is established prior to the commencement of the test and must comply with the following conditions: \(0<x_{1}<x_{2}<\cdots<x_{m}\) and \(n=R_{1}+\cdots+R_{m}+m\). Progressive Type-II censoring provides both practicality and flexibility by permitting units to be removed at any phase following a failure. The notation \(R=(R_{1},\ldots,R_{m})\) denotes the censoring scheme. Here, \(X=(X_{1},\ldots,X_{i},\)\(\ldots,X_{m})\) represents the corresponding progressive censored samples derived from the generalized Pareto distribution. A random variable \(X\) is said to follow a generalized Pareto distribution if its probability density function (PDF) and cumulative distribution function (CDF) are defined as follows:\[f(x;\alpha,\lambda)=\lambda\alpha(1+\lambda x)^{-(\alpha+1)},x>0, \tag{1}\] \[F(x;\alpha,\lambda)=1-(1+\lambda x)^{-\alpha},x>0, \tag{2}\]  
where the parameters associated with the scale and shape are denoted as \(\lambda>0\) and \(\alpha>0\). We refer to this distribution as \(GPD(\alpha,\lambda)\). It is also recognized as the second kind of Pareto distribution or the Lomax distribution. The likelihood function for samples obtained from the progressive Type-II censoring scheme is as follows:  
\[l(\alpha,\lambda|\hat{x})=\alpha^{m}\lambda^{m}\prod_{i=1}^{m}(\lambda x_{i}+ 1)^{-(\alpha+1)}[(\lambda x_{i}+1)^{-\alpha}]^{R_{i}}, \tag{3}\]  
where \(\hat{x}=(x_{1},\ldots,x_{i},\ldots,x_{m})\) represents the observed value of \(X=(X_{1},\ldots,X_{i},\ldots,X_{m})\). Consequently, we can derive the log-likelihood function as follows:  
\[L(\alpha,\lambda|\hat{x})=m\ln(\alpha\lambda)-(\alpha+1)\sum_{i=1}^{m}\ln( \lambda x_{i}+1)-\alpha\sum_{i=1}^{m}R_{i}\ln(\lambda x_{i}+1), \tag{4}\]  
\[\frac{\partial L}{\partial\alpha}=\frac{m}{\alpha}-\sum_{i=1}^{m}(1+R_{i})\ln (\lambda x_{i}+1)=0, \tag{5}\]  
\[\frac{\partial L}{\partial\lambda}=\frac{m}{\lambda}-(\alpha+1)\sum_{i=1}^{m} \frac{x_{i}}{\lambda x_{i}+1}-\alpha\sum_{i=1}^{m}\frac{R_{i}x_{i}}{\lambda x_ {i}+1}=0. \tag{6}\]  
To obtain maximum likelihood estimates, the traditional approach is the Newton-Raphson method, as referenced in [17]. However, a significant challenge with this method is the need to compute the quadratic partial derivatives of the log-likelihood function. In the context of this censored model, determining the roots of Equations (5) and (6) can often be quite complex. Therefore, the expectation-maximization algorithm, as introduced in [18], is employed to address this issue and facilitate the calculation of MLEs. The EM algorithm ensures convergence to a stable point, requiring only the pseudo log-likelihood function of the complete sample to reach its maximum value, thereby effectively resolving the aforementioned challenges. The EM algorithm consists of two main steps. The E-step entails calculating the expectation of the log-likelihood function using censored data based on the current parameter estimates. The M-step involves maximizing the log-likelihood function to obtain updated parameter estimates. Let \(Z=(Z_{1},\ldots,Z_{r},\ldots,Z_{m})\) represent the censored samples. Each \(Z_{i}\) is a \(1\times R_{i}\) vector recorded as \(Z_{i}=\big{(}Z_{i1},Z_{i2},\ldots,Z_{iR_{i}}\big{)}\), where \(i=1,2,\ldots,m\). The variable \(X=(X_{1},\ldots,X_{r},\ldots,X_{m})\) denotes the observed sample, and \(Z_{i}\) signifies the censored data following the failure of \(X_{i}\). We define \(Q=(X,Z)\) to represent the complete sample.