The most general definition of the Likelihood function for continuous data (including truncation and censoring)

Question

How would you rigorously define the likelihood function for censored/truncated observations? Even in most lifetime/reliability literature (where these types of observations are frequently encountered) it is often defined (for continuous random variables) to be

$$\mathcal{L}(\theta;\mathbf{x}) = \prod_i f(x_i;\theta)$$

This does not include censored and truncated (although density $f$ could be slightly altered) observations however (I know what the likelihood function will look like, I'm just not sure how to neatly define it from a mathematical point of view).

You also frequently encounter the following definition

$$\mathcal{L}(\theta;\mathbf{x}) = \prod_i P(x_i;\theta)$$

This does not apply to continuous data however (What is the difference between "likelihood" and "probability"?).

I'm looking for something similar to this: Maximum Likelihood Estimation where the likelihood function is defined as follows

Suppose one has, for an observation $X$ with distribution $P_{\theta}$ depending on an unknown parameter $\theta \in \Theta \in \mathbb{R}^k$, the task to estimate $\theta$. Assuming that all measures $P_{\theta}$ are absolutely continuous relative to a common measure $\nu$, the likelihood function is defined by

$$\mathcal{L}(\theta;X) = \frac{dP_{\theta}}{d\nu}(X)$$

It was quite some time since I studied measure theory, so I'm not certain what the truncation/censoring would correspond to w.r.t. the measure and distribution (if even applicable).

Answer 1

Klein and Moeschberger present expressions for likelihoods under censoring or truncation in "Survival Analysis: Techniques for Censored and Truncated Data" (Springer; 2nd edition, 2003). The formulas represent the following general principles (page 74)

An observation corresponding to an exact event time provides information on the probability that the event’s occurring at this time ... For a right-censored observation all we know is that the event time is larger than this time, so the information is the survival function evaluated at the on study time. Similarly for a left-censored observation, all we know is that the event has already occurred, so the contribution to the likelihood is the cumulative distribution function evaluated at the on study time. Finally, for interval-censored data we know only that the event occurred within the interval, so the information is the probability that the event time is in this interval. For truncated data these probabilities are replaced by the appropriate conditional probabilities.

With independence of event and censoring/truncation times, likelihoods $L$ are proportional to the following for different types of observations, where $f(x)$ is the probability distribution of lifetimes and $S(x)$ is the corresponding survival function (page 74):

• exact lifetime, $L \propto f(x)$, where $x$ is the observed event time;

• right-censored, $L \propto S(C_r)$, where $C_r$ is the right-censoring time;

• left-censored, $L \propto 1- S(C_l)$, where $C_l$ is the left-censoring time;

• interval-censored, $L \propto [S(L)-S(R)]$, where $L$ and $R$ are the left- and right-censoring times of the interval;

• left-truncated, $L \propto f(x)/S(Y_L)$, where $x$ is the observed event time and $Y_L$ is the left-truncation time;

• right-truncated, $L \propto f(x)/[1-S(Y_R)]$, where $x$ is the observed event time and $Y_R$ is the right-truncation time;

• interval-truncated, $L \propto f(x)/[S(Y_L)-S(Y_R)]$, where $x$ is the observed event time and $Y_L$ and $Y_R$ are the left- and right-truncation times.

The overall expression for the likelihood (under appropriate assumptions about independence) is the product of terms associated with each instance of each type of censoring/truncation. Note that for regression modeling $f(x)$ would be a function of covariate values associated with an individual.

See this page for an introduction to implementation.