Likelihood of censored data Let $X_1,X_2,\ldots, X_{n_1}$ be IID with PDF $f(x-\theta) $, for $-\infty<x<\infty$ and $-\infty<\theta<\infty$.  Denote the CDF of $X_i$ by $F(x-\theta)$.
Let $Z_1,Z_2, \ldots, Z_{n_2}$ denote the censored observations. For these observations we only know that $Z_j >a $ for some $a$ that is known and that the $Z_j$s are independent of the $X_i$s. 
Then the "observed" and "complete" , that is for both the observed and the censored data, likelihoods are given by:
$$L(\theta| \mathbf{x} )= [1-F(a-\theta)]^{n_2} \prod_{i=1}^{n_1} f(x_i -\theta) $$
$$ L( \theta |\mathbf{x,z})=\prod^{n_1}_{i=1}f(x_i -\theta) \prod_{i=1}^{n_2}f(z_i-\theta) $$
My question is why do the likelihoods take that form? I believe for the $Z_j$s we have a left censoring as they only take greater than $a$ values, as opposed to the $X_is$ that are not restricted in that way. We also know that they are mutually independent. 
Could it be that the "observed" likelihood is obtained after using that independence? We have $n_2$  observations greater than $a$, times the joint density of the $X_i$s, right?. 
A little more explanation on these two equations would go a long way. 
Thank you in advance.
 A: In the case of your first equation:
$$L(\theta| \mathbf{x} )= [1-F(a-\theta)]^{n_2} \prod_{i=1}^{n_1} f(x_i -\theta) $$
it's assumed you only observe the $n_1$ values of $\mathbf{x}$, but that you also know there were $n_2$ observations censored at $a$.  It might have been better for the authors to have written the left hand side as $L(\theta | \mathbf{x}, n_2, a)$ for clarity.  
If you didn't know anything about the censored observations, including how many of them there were, the likelihood function would then be a function solely of the $\mathbf{x}$, by necessity.  You'd then get:
$$L(\theta| \mathbf{x} )=\prod_{i=1}^{n_1} f(x_i -\theta) $$
which is what I suspect you expected to see for $L(\theta | \mathbf{x})$.  
In the case of your second equation:
$$L( \theta |\mathbf{x,z})=\prod^{n_1}_{i=1}f(x_i -\theta) \prod_{i=1}^{n_2}f(z_i-\theta) $$
it's assumed that you are observing all the $\mathbf{z}$ values, as in, none of them are censored.  Hence the likelihood function is the product of the likelihood functions for the $\mathbf{x}$ and the $\mathbf{z}$, and the label "complete data log likelihood".
