Censoring

Study guides: Exam FAM, Exam MAS-I

Censoring occurs when the exact value of an observation is unknown, but it is known to exceed (or fall below) a threshold. In actuarial and survival analysis, right-censoring is most common: a claim or lifetime is observed only up to some censoring time $c$ , after which observation ends.

Right-censored likelihood contribution: $\text{If } X > c: \quad L_i(\theta) = P(X > c \mid \theta)$

$= S(c \mid \theta)$

Observed (uncensored) likelihood contribution: $\text{If } X = x \leq c: \quad L_i(\theta) = f(x \mid \theta)$

Right-censoring: true value exceeds $c$ (e.g., policy limit reached, study ends before claim settles)
Left-censoring: true value is below a detection threshold
Interval censoring: the value is known only to lie within an interval $(a, b)$
Right-censoring must be accounted for in Maximum Likelihood Estimation: a right-censored record is $\min(X, c) \le X$ , so treating it as an exact value causes downward bias in estimated means or survival times
An observation that is not censored (the exact value is known) contributes $f(x|\theta)$ to the likelihood

MLE With Censored Exponential Data Three claims have observed values $x_1 = 2$ , $x_2 = 3$ , and one claim is right-censored at $c = 4$ (i.e., we know $X_3 > 4$ ). Assuming $X \sim \text{Exp}(\theta)$ , write the likelihood.

Answer $L(\theta) = f(2|\theta)\cdot f(3|\theta)\cdot S(4|\theta) = \frac{1}{\theta}e^{-2/\theta}\cdot\frac{1}{\theta}e^{-3/\theta}\cdot e^{-4/\theta} = \frac{1}{\theta^2}e^{-9/\theta}$ Taking the log and differentiating: $\ell(\theta) = -2\ln\theta - 9/\theta$ ; $\ell'(\theta) = -2/\theta + 9/\theta^2 = 0$ gives $\hat{\theta} = 9/2 = 4.5$ . Note: the censored observation still contributes information (the $e^{-4/\theta}$ term).