## 2 The Classical Estimators of the Parameters and Reliability
### Uniformly Minimum-Variance Unbiased Estimation  
Suppose that \(\underline{X}=(X_{1},X_{2},\cdots,X_{n})^{\prime}\) represents a random sample of the Burr XII distribution. Then, the statistic \(S=\sum_{i=1}^{n}\ln(1+X_{i}^{\alpha})\) has a \(Gamma\left(n,\frac{1}{\beta}\right)\) distribution, for which the PDF is structured as follows:  
\[d(x;n,\beta)=\frac{\beta^{n}}{\Gamma(n)}x^{n-1}e^{-\beta x},x>0,\]  
where \(\Gamma(\cdot)\) denotes the Gamma function. The statistic \(S\) can be proved to be a sufficiently complete statistic for \(\beta\). According to the Lehmann-Scheffe theorem, given \(E\left(\frac{1}{S}\right)=\frac{\beta}{n-1}\), the UMVUE of the parameter \(\beta\) is  
\[\hat{\beta}_{UMVUE}=\frac{n-1}{S}, \tag{11}\]  
and the approximate UMVUE expression of \(R(t)\) is  
\[\bar{R}_{UMVUE}(t)\cong(1+t^{\alpha})^{\hat{\beta}_{UMVUE}}. \tag{12}\]  
In order to obtain an accurate expression for the UMVUE of \(R(t)\), we present the following theorems. **Theorem 1**. : _When considering the random sample \(\underline{X}=(X_{1},X_{2},\cdots,X_{n})^{\prime}\) from the Burr XII distribution, the UMVUE of the PDF \(f(x)\) at a given point, \(x\), is expressed as_  
\[\hat{f}_{UMVUE}(x)= \begin{cases}\frac{\alpha x^{n-1}(n-1)\left(1-s^{-1}\ln(1+x ^{\alpha})\right)^{n-2}}{s(1+x^{\alpha})},&s>\ln(1+x^{\alpha})\\ 0,&o.w.\end{cases}  \tag{13}\]  
_where \(s=\sum_{i=1}^{n}\ln(1+x_{i}^{\alpha})\)._  
Proof. : Based on the Lehmann-Scheffe theorem, as \(S\) is a completely sufficient statistic, we can infer that \(f(X_{n}|S)\), which is the conditional PDF of \(X_{n}\) given \(S\), is the UMVUE of \(f(x)\). Let \(L\) denote \(\sum_{i=1}^{n-1}\ln(1+X_{i}^{\alpha})\), for which the PDF is structured as  
\[\varphi(l)=\frac{\beta^{n-1}}{\Gamma(n-1)}l^{n-2}e^{-\beta l},l>0,\]  
where \(l=\sum_{i=1}^{n-1}\ln(1+x_{i}^{\alpha})\). Then, considering the independence between \(L\) and \(X_{n}\), the joint PDF of \(L\) and \(X_{n}\) is written as \(\varphi(l,x_{n})=\varphi(l)\cdot f(x_{n})\). As \(S=L+\ln(1+X_{n}^{\alpha})\), we perform a variable transformation to obtain the joint PDF of \((S,X_{n})\), which is  
\[\varphi(s,x_{n})=\frac{\beta^{n-1}}{\Gamma(n-1)}(s-\ln(1+x_{n}^{\alpha}))^{n- 2}e^{-\beta(s-\ln(1+x_{n}^{\alpha}))}\cdot\alpha\beta x_{n}^{\alpha-1}(1+x_{ n}^{\alpha})^{-\beta-1},s>\ln(1+x_{n}^{\alpha}).\]  
Additionally, the PDF of \(S\) is  
\[g(s)=\frac{\beta^{n}}{\Gamma(n)}s^{n-1}e^{-\beta s}.\]Thus, the conditional PDF of \(X_{n}\) given \(S\) is obtained as \(\frac{\varphi(s,x_{n})}{g(s)}\). That is, the UMVUE of \(f(x)\) is calculated as  
\[\hat{f}_{UMVUE}(x_{n})=\frac{\alpha x_{n}^{\alpha-1}(n-1)\big{(}1-s^{-1}\ln(1+x _{n}^{\alpha})\big{)}^{n-2}}{s(1+x_{n}^{\alpha})},s>\ln(1+x_{n}^{\alpha}).\]  
Based on Theorem 1, the UMVUE of \(R(t)\) is presented in Theorem 2. **Theorem 2**. : _Given the estimate \(\hat{f}_{UMVUE}(x)\) of the PDF \(f(x)\), the UMVUE of the reliability function \(R(t)\) is derived as_  
\[\hat{R}_{UMVUE}(t)=\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx=\Big{(}1-s^{-1}\ln(1+t ^{\alpha})\Big{)}^{n-1},s>\ln(1+t^{\alpha}). \tag{14}\]  
Proof. : In order to prove the first equality of the theorem, we note that \(\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx\) is the function of \(S\), which is sufficient and complete for \(\beta\). Thus, we only need to prove that \(\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx\) is unbiased for \(R(t)\). \[E\Big{(}\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx\Big{)} =\int_{0}^{\infty}\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx\cdot g(s)ds\] \[=\int_{t}^{\infty}\int_{0}^{\infty}\frac{\varphi(s,x_{n})}{g(s)} g(s)dsdx\] \[=\int_{t}^{\infty}f(x_{n})dx\] \[=R(t),\]  
where \(g(s)\) denotes the PDF of \(S\). Then, we can calculate the specific expression for \(\hat{R}_{UMVUE}(t)\) as follows:  
\[\hat{R}_{UMVUE}(t) =\int_{t}^{\infty}\hat{f}_{UMVUE}(x)dx\] \[= \begin{cases}\int_{t}^{\infty}\frac{\alpha x^{\alpha-1}(n-1) \big{(}1-s^{-1}\ln(1+x^{\alpha})\big{)}^{n-2}}{s(1+x^{\alpha})}dx,&s>\ln(1+t^{ \alpha})\\ 0,&o.w.\end{cases} \] \[=