Type 2 Censoring

Definition

Type 2 Censoring

when $r=3$

Suppose that $T_1,T_2,\dots ,T_n$ are i.i.d. random variables represent survival times with CDF $F$, and $C_1,C_2,\dots ,C_n$ are the censoring times. And let ${\delta }_i=\{\begin{array}{ll}1& T_i\le C_i\text{: uncensored}\\ 0& T_i>C_i\text{: censored}\end{array}$ be the censoring indicator. We observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$.

In a type 2 censoring setting, we observe first $r$ out of $n$ experiment $T_1,\dots ,T_r$. In other words, for the order statistics $T_{(i)}$ of $T_i$, we only observe $T_{(1)}\le T_{(2)}\le \cdots \le T_{(r)}$. Where $C_1=C_2=\cdots =C_n$ are not constants, but random variables.

Likelihood of Type 2 Censoring Data

The likelihood function of type 2 censored data can be computed using the same equation used for type 1 censored data but computing the joint-PDF of the Order Statistic $T_{(1)}\le \cdots \le T_{(r)}$ is easier.

The joint PDF of $T_{(1)}\le \cdots \le T_{(r)}$ is derived as $L(\theta )=\frac{n!}{(n-r)!}\left[\prod _{i=1}^{r}f(t_{(i)})\right]S(t_{(r)}{)}^{n-r}$