# The Combined-Unified Hybrid Censored Samples from Pareto Distribution: Estimation and Properties
## 2 Likelihood Function of C-UHCS
### Case 3: \(k\) and \(\alpha\) Are Unknown  
The MLEs of \(k,\alpha,R(t)\) and \(H(t)\) can be derived, respectively, by  
\[\tilde{k}=x_{1:n},\;\tilde{\alpha}=\frac{r}{\log(y_{2})},\;\tilde{R}(t)=exp \left(\frac{r\log\left(\frac{x_{1:n}}{t}\right)}{\log[y_{2}]}\right)\text{ and }\tilde{H}(t)=\frac{r}{t\log[y_{2}]}, \tag{21}\]  
where  
\[y_{2}=\left(\frac{t}{x_{1:n}}\right)^{n-r}\prod_{i=2}^{r}x_{i:n}. \tag{22}\]  
From (21), we see that \((\tilde{k},\tilde{\alpha})\) are jointly sufficient statistics for \((k,\alpha)\). The following theorem states the mean and variance of the MLE of \(\frac{1}{\tilde{\kappa}}\) in this case. **Theorem 2**. : _If \(X_{1:n},X_{2:n},\ldots X_{r:n}\) be the C-UHCS\((m,l;t_{1},t_{2})\) from Pareto distribution given in (1). For the biased estimate of \(\frac{1}{\tilde{\kappa}}\), we have_  
\[E\left(\frac{1}{\tilde{\kappa}}\right)=\frac{r-1}{r\alpha},\;Var(\frac{1}{ \tilde{\kappa}})=\frac{r-1}{r^{2}\alpha^{2}},\text{ and }MSE(\frac{1}{\tilde{\kappa}})=\frac{1}{r \alpha^{2}}, \tag{23}\]  
_while \(\log[y_{2}]/(r-1)\) is an unbiased estimate of \(1/\alpha\) with variance \(1/((r-1)\alpha^{2})\) and relative efficiency \(1-1/r\)._  
Proof. : See Appendix B. **Theorem 3**. : _If \(X_{1:n},X_{2:n},\ldots X_{r:n}\) be the C-UHCS\((m,l;t_{1},t_{2})\) from Pareto distribution given in (1). For the biased estimate of \(\alpha\), we have_  
\[E(\tilde{\kappa}) = \frac{r}{r-2}\alpha,\;r>2, \tag{24}\] \[Var(\tilde{\alpha}) = \frac{r^{2}\alpha^{2}}{(r-3)(r-2)^{2}},\;r>3,\] (25) \[MSE(\tilde{\kappa}) = \frac{r^{2}+2r-6}{(r-3)(r-2)^{2}}\alpha^{2},\;r>3,. \tag{26}\]  
_and for the unbiased estimate of \(\alpha\), we have_  
\[\hat{\alpha}=\frac{r-2}{r}\tilde{\kappa}=\frac{r-2}{\log y_{2}},\;Var(\hat{ \alpha})=\frac{\alpha^{2}}{r-3},\;r>3\;and\;RE(\hat{\alpha})=\frac{r-3}{r}. \tag{27}\]Proof. : See Appendix C.  
**Theorem 4**. : _If \(X_{1:n},X_{2:n},\ldots X_{r:n}\) be the C-UHCS\((m,l;t_{1},t_{2})\) from Pareto distribution given in (1). Then the unbiased MLE of k of Pareto distribution is_  
\[\left(1-\frac{\log y_{2}}{n(r-1)}\right)X_{1:n}. \tag{28}\]  
Proof. : See Appendix D.  
**Remark 1**. : _The MLEs of Pareto parameters and their properties based on complete sample given in Baxter [29] can be easily derived from our results in the cases 1, 2, and 3 by setting \(r=n\)._  
Now, we apply the normality appromimation of the MLEs to obtain the approximate confidence intervals for \(k\) and \(\alpha\). The variance-covariance matrix of the parameters \(\hat{V}=[\sigma_{i,j}],i,j=1,2\) can be written as  
\[V(\hat{k},\hat{\alpha})=-\left[ \begin{array}{cc}\frac{\partial^{2}\text{log }[L]}{\partial k^{2}}&\frac{\partial^{2}\text{log}[L]}{\partial k\partial \alpha}\\ \\ \frac{\partial^{2}\text{log}[L]}{\partial\alpha\partial k}&\frac{\partial^{2} \text{log}[L]}{\partial\alpha^{2}}\end{array} \right]^{-1}_{(k=\hat{k},\alpha =\hat{k})}, \tag{29}\]  
where the elements of the observed variance-covariance matrix can be derived from (10) as  
\[\frac{\partial^{2}\log[L]}{\partial k^{2}}=\frac{-n\alpha}{k^{2}},\qquad\frac {\partial^{2}\log[L]}{\partial\alpha^{2}}=\frac{-r}{\alpha^{2}},\qquad\frac{ \partial^{2}\log[L]}{\partial\alpha\partial k}=\frac{n}{k}, \tag{30}\]  
and hence the minimum variance bound of the MLEs of \(\alpha\) and \(\frac{1}{\alpha}\) are given, respectively, by \(\frac{\alpha^{2}}{r}\) and \(\frac{1}{n\alpha^{2}}\). \(V(k,\alpha)\) takes the form  
\[V(k,\alpha)=\frac{k^{2}\alpha}{n(r-n\alpha)}\left[ \begin{array}{cc}\frac{ r}{\alpha^{2}}&\frac{k}{n}\\ \\ \frac{k}{n}&\frac{n\alpha}{k^{2}}\end{array}