Fitting distributions on censored data

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/

https://stats.stackexchange.com/questions/49443/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.

• This question would get the attention of more experts if posted on CrossValidated.stackexchange. – David G. Stork Jul 5 '18 at 22:38

Both of the links you provided assume that the data comes from a continuous distribution, ie try to fit a continuous distribution. For the censored data I guess you are saying that $F (x)$ is used in the likelyhood product expression (to maximise) instead of the product of 'discrete terms' which we don't want to use, for the data points bellow the chosen threshold (similary for above some threshold).