Left Censoring

Definition

Suppose that $T_1,T_2,\dots ,T_n$ are i.i.d. random variables represent survival times, and $C_1,C_2,\dots ,C_n$ are the censoring times, which can be both random variable or constant. And let ${\delta }_i=\{\begin{array}{ll}1& T_i\ge C_i\text{: uncensored}\\ 0& T_i<C_i\text{: censored}\end{array}$ be the censoring indicator.

In a left censoring setting, we observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\max (T_i,C_i)$. In other words the event has already occurred before it becomes the subject of observed.