# The Combined-Unified Hybrid Censored Samples from Pareto Distribution: Estimation and Properties
## 2 Likelihood Function of C-UHCS  
Consider \(X_{1:n},X_{2:n},\ldots X_{r:n}\) are the lifetimes of units that placed on a life-test, and let cumulative distribution (cdf) \(F(x)\) and probability density distribution (pdf) \(f(x)\) and assume that, for any case, the experiment is terminated at \(t\) that may refer to time \(t_{1}\), \(t_{2}\), observation \(x_{m:n}\) or observation \(x_{l:n}\), and let \(r\) denote the maximum number of failures until \(t\) equal, respectively, \(D_{1},D_{2},m\) and \(l\). [PERSON] and [PERSON] [28] have proposed the likelihood function under the censoring samples C-UHCS\((m,l;t_{1},t_{2})\) under different choices of \(r\), \(t\) and \(\mathbf{x}_{r:n}=(x_{1:n},x_{2:n},\ldots,x_{r:n})\) as  
\[L(\Omega|\mathbf{x}_{r:n})=\frac{n!}{(n-r)!}[1-F(t)]^{n-r}\prod_{i=1}^{r}f(x_ {i:n}), \tag{6}\]  
where \(r\) and \(t\) can be chosen in the different cases of censoring as:  
\begin{tabular}{l l l l l} \hline \hline  & & \(\boldsymbol{L^{(C)}(\Omega|\mathbf{x})}\) & & \(\boldsymbol{L^{(U)}(\Omega|\mathbf{x})}\) \\ \hline
**Cases** & \(\boldsymbol{r}\) & \(\boldsymbol{t}\) & \(\boldsymbol{r}\) & \(\boldsymbol{t}\) \\ \hline
1: \(0<t_{1}<X_{m:n}<t_{2}<X_{l:n}\) & \(m\) & \(X_{m:n}\) & \(D_{2}\) & \(t_{2}\) \\
2: \(0<t_{1}<X_{m:n}<X_{l:n}<t_{2}\) & \(m\) & \(X_{m:n}\) & \(l\) & \(X_{l:n}\) \\
3: \(0<t_{1}<t_{2}<X_{m:n}<X_{l:n}\) & \(D_{2}\) & \(t_{2}\) & \(m\) & \(X_{m:n}\) \\
4: \(0<X_{m:n}<X_{l:n}<t_{1}<t_{2}\) & \(l\) & \(X_{l:n}\) & \(D_{1}\) & \(t_{1}\) \\
5: \(0<X_{m:n}<t_{1}<X_{l:n}<t_{2}\) & \(D_{1}\) & \(t_{1}\) & \(l\) & \(X_{l:n}\) \\
6: \(0<X_{m:n}<t_{1}<t_{2}<X_{l:n}\) & \(D_{1}\) & \(t_{1}\) & \(D_{2}\) & \(t_{2}\) \\ \hline \hline \end{tabular}  
\(L^{(C)}(\Omega|\mathbf{x})\) is the likelihood corresponding to CHCS\((m,l;t_{1},t_{2})\) given by [PERSON] and [PERSON] [26] as  
\[L^{(C)}(\Omega|\mathbf{x})=\left\{\begin{array}{ll}\dfrac{n!}{(n-m)!}[1-F(x_{m: n})]^{n-m}\prod\limits_{i=1}^{m}f(x_{i:n});&D_{1}=0,\cdots,m-1,D_{2}=m,\\ \dfrac{n!}{(n-D_{2})!}[1-F(t_{2})]^{n-D_{2}}\prod\limits_{i=1}^{D_{2}}f(x_{i:n} );&D_{1},D_{2}=0,\cdots,m-1,\\ \dfrac{n!}{(n-l)!}[1-F(x_{l:n})]^{n-l}\prod\limits_{i=1}^{l}f(x_{i:n} );&D_{1}=D_{2}=l,\\ \dfrac{n!}{(n-D_{1})!}[1-F(t_{1})]^{n-D_{1}}\prod\limits_{i=1}^{D_{1}}f(x_{i:n} );&D_{1}=D_{2}=m,\cdots,l-1,\end{array}\right. \tag{7}\]  
and \(L^{(U)}(\Omega|\mathbf{x})\) is the likelihood corresponding to UHCS\((m,l;t_{1},t_{2})\) given by [PERSON] et al. [27] as  
\[L^{(U)}(\Omega|\mathbf{x})=\left\{\begin{array}{ll}\dfrac{n!}{(n-D)!}[1-F(t _{1})]^{n-D}\prod\limits_{i=1}^{m}f(x_{i:n});&D_{1}=D_{2}=D=l,\cdots,n,\\ \dfrac{n!}{(n-l)!}[1-F(x_{l:n})]^{n-l}\prod\limits_{i=1}^{l}f(x_{i:n} );&D_{1}=m,\cdots,l-1,D_{2}=l,\\ \dfrac{n!}{(n-D_{2})!}[1-F(t_{2})]^{n-D_{2}}\prod\limits_{i=1}^{D_{2}}f(x_{i} );&D_{1},D_{2}=m,\cdots,l-1,\\ \dfrac{n!}{(n-l)!}[1-F(x_{l:n})]^{n-l}\prod\limits_{i=1}^{l}f(x_{i:n} );&D_{1}=0,\cdots,m-1,D_{2}=l,\\ \dfrac{n!}{(n-D_{2})!}[1-F(t_{2})]^{n-D_{2}}\prod\limits_{i=1}^{D_{2}}f(x_{i:n });&D_{1}=0..m-1,D_{2}=m...l-1,\\ \dfrac{n!}{(n-m)!}[1-F(x_{m:n})]^{n-m}\prod\limits_{i=1}^{m}f(x_{i:n} );&D_{1},D_{2}=0,\cdots,m-1.\end{array}\right. \tag{8}\]  
**3. The Maximum Likelihood Estimates**  
Let \(X_{1:n},X_{2:n},\ldots X_{r:n}\) be the C-UHCS\((m,l;t_{1},t_{2})\) from the Pareto distribution given in (1). The likelihood function given in (8) in this case may written as  
\[L=\dfrac{n!}{(n-r)!}\left[\left(\dfrac{k}{t}\right)^{n-r}\prod\limits_{i=1}^ {r}\dfrac{\alpha}{x_{i:n}}\left(\dfrac{k}{x_{i:n}}\right)^{\alpha},\;\;k\leq x _{1:n}\leq x_{2:n}\leq\cdots\leq t, \tag{9}\]  
and hence  
\[\log L=\log\left[\dfrac{n!}{(n-r)!\prod\limits_{i=1}^{r}x_{i:n}}\right]+r \log\alpha+n\alpha\log k-\alpha\log\left(t^{n-r}\prod\limits_{i=1}^{r}x_{i:n} \right). \tag{10}\]  
From (9), we consider the following cases: