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.

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    $\begingroup$ I do not understand the questions in your comment. By definition, the likelihood of $(x,z)$ is the probability of observing $(x,z)$ as a function of $\theta$. The information you give in your first paragraph more or less stipulates how to find the probability of any individual $x_i$ or $z_j$ and the independence assumption implies those probabilities are multiplied: that's all that's going on here. For an example, you might like to read over my answer at stats.stackexchange.com/a/49456/919. $\endgroup$
    – whuber
    Jan 9 '14 at 20:18
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    $\begingroup$ @whuber I guess I do not understand why we have the probability that $Z$ is greater than $a$ in the likelihood of $x$. I am having trouble understanding the presence of the first term in the product, based on the information I have given. $\endgroup$
    – JohnK
    Jan 9 '14 at 20:24
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    $\begingroup$ The first equation makes no sense. Before "becoming" a likelihood, it was a joint density, $f(\mathbf x \mid \theta) = \prod_{i=1}^{n_1} f(x_i -\theta)$. The $\mathbf z$'s must somehow appear in the argument of $L()$ in order for anything related to them to be included in the RHS. $\endgroup$ Jan 9 '14 at 20:31
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    $\begingroup$ @Alecos although the description of notation is a little off in the post, its meaning is clear: to understand the censoring, we contemplate the (hypothetical) uncensored data (giving the second likelihood, where the $z_i$ are uncensored observations governed by the distribution $\theta$). After censoring, the likelihood equals the first expression. The indexing and naming of variables may be a little problematic here; I prefer to use unique indexes for the variables, as illustrated at stats.stackexchange.com/a/49456, because it helps avoid confusion. $\endgroup$
    – whuber
    Jan 9 '14 at 21:19
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    $\begingroup$ John, I just did, and you have misinterpreted the book (or you have transcribed the information here in a way that is confusing, at least to me -and I am not referring to the likelihoods). Later I will write an answer on the issue. (cc @whuber) $\endgroup$ Jan 9 '14 at 21:21

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".


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