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).


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