Fitting distributions on censored data

Question

My question deals with fitting distributions on censored data; for the purposes of clarity, we can consider a continuous distribution which is both left and right-censored. In such a case, the variates are "clubbed" into a maximal upper-value if they are greater than a threshold $t_h$, or into a minimal lower-value if they are less than a threshold $t_l$. We have only observed this censored data and want to fit a distribution which in some sense appropriately models the data which we have observed.

I've seen a number of cases in dealing with fitting distributions, where parameter fitting is done via Maximum Likelihood Estimation. In such cases, the probability for the points which are censored is set via the CDF F(x) = P(X < x): $F(x)$ is used to evaluate left-truncated points and $1-F(x)$ is used to evaluate right-truncated points. Meanwhile, for the non-truncated points, the probability density f(x) is used for evaluation. For some examples, please see these posts if you don't understand what I mean:

https://www.r-bloggers.com/fitting-censored-log-normal-data/

How to model this odd-shaped distribution (almost a reverse-J)

My question is, why is it commonly accepted to fit the censored parts using probability $mass$, but the non-censored parts using probability $density$? Since these are different units, don't the results become unstable or influenced by the differences in scales of magnitude differences that might exist between the density and mass?

My rationalization of why this procedure might be okay is that in a model-selection regime such as distribution fitting, these problems persist across various parameters of the model class -- in some sense, we have a "level playing field" across contender distributions. This doesn't really address the problem of different scales for mass and density, but at least it seems "fair."

Could someone shed some light on this? Any other pointers on dealing with such distributions (continuous over a range, and then with point masses thrown in) would be helpful as I'm very new to this space.

Answer 1

A short answer which might be expanded: This is really about how the likelihood function is defined, see for instance How to rigorously define the likelihood?. It doesn't really matter if you use probability densities or probability mass functions to define your likelihood. The likelihood function is always defined with respect to some dominating measure on the sample space of the observations.

If the dominating measure is Lebesgue measure on the real line, we get probability distributions defined via densities. If the dominating measure is counting measure, we get probability mass functions. But there are many other possibilities, an example is modeling daily rainfall, where there might be some positive probability of zero rain, but else a density on positive values. That can be represented as a density (formally: Radon-Nikodym derivative) with respect to a sum of Lebesgue measure on the positive real line, and a probability atom at zero.

In your case, with censoring at the high and low ends, say at $a < b$, your dominating measure is a sum of Lebesgue measure on $(a,b)$ and probability atoms at $a$ and at $b$. In this abstract setting, this is no different than a density with respect to Lebesgue measure or with respect to counting measure. So you have nothing to be preoccupied about! What is important is that the dominating measure is the same for all the possible particular models (whether parametrized or not) that you entertain. And, as far as I know, this framework do not allow for estimating the dominating measure, that has to be known by the modeler.

For information about Radon-Nikodym derivatives and dominating measures, see Interpretation of Radon-Nikodym derivative between probability measures?