The (right-) censoring of likelihood function is defined as:

$$ L(\theta) = \prod_{i=1}^m f(x_i;\theta) \cdot \prod_{i=m+1}^n P_{\theta}(X_j>a) $$

where observations $x_i$, $i=1,...,m$ have known values and for $m+1,...,n$ the values are unknown.

I would like to know, how has this been derived? I.e. why is this the definition for censoring?

  • $\begingroup$ @BCLC If the observation $x_i$ (its value) is higher than $a$ then it's not observed (it's censored). Therefore there's the probability of $P_{\theta}(X_j>a)$, in other words, the probability of an unobserved observation. $\endgroup$ – mavavilj Mar 19 '16 at 16:44

What you have shown is the likelihood function when right-censoring is present. There is also left-censoring, interval censoring, and truncation. These would introduce similar terms as your last one.

An intuitive way to think about it is to look at the discrete case, where likelihood is probability. So imagine a man rolling a die until he gets a six (geometric random variable). He runs the experiment twice and tells you it took 4 rolls the first time and some number greater than or equal to 9 the second time.

What is the probability of this occurring if the chance $p$ of rolling a six is constant for each roll?

$$P[X_1=4, X_2 \ge 9] = \left[ \left( 1-p \right)^3 p \right] \left[ \left( 1-p \right)^8 \right] $$

Now, extending to the continuous case, the observed values will use the density function and the right-cesored values will use the complement of the cumulative distribution function.

This is an informal answer. For a rigorous look, you'll need to find a good reference/textbook.

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