Censoring and truncation -highlighting the differences

Survival data are very common in medical science, actuarial science, astronomy, demography, and many other scientific areas. The most typical characteristic of survival data is incompleteness, where by far the most common models are those of censoring and truncation. Although quite different in nature, the left truncation and the right censoring models result in nonparametric estimates which are very similar in form. However, in other models of truncation and censoring this similarity breaks and estimation should be based on the special properties of each type of incompleteness. This note contrasts two simple models, one of censoring with a known censoring distribution and the other of truncation with a known truncation distribution. The goal is to highlight the differences between the two types of incomplete data in a way that can help analyze and interpret more complicated survival data.