# 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/

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.

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.
• Thanks for this! This provides some formal justification for the informal intuition that I had about this case. One question: in such a dominating measure, how might one compare intuitively the "relative probability" of a sample in the Lebesgue region (a,b) and atoms at $a$ and $b$? Perhaps it's nonsensical or an ill-defined question given P(X = x) = 0 on (a, b). I'm interested from the perspective of comparing likelihood of a variate observed on (a,b) versus at $a$ or $b$, if that makes sense. Thank you so much for your help! Can one naively compare these two likelihoods? Jul 6, 2018 at 17:53
• You never needs such a comparison. For each data point $x_i$, it will be either $a$, $b$ or in $(a,b)$. What you compare is different parameter values, that is, different models, at the same data points! Jul 6, 2018 at 18:32