Random Censoring

Definition

Random Censoring

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, which can be both random variable or constant. 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.

In a random censoring setting, we only observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$, and the censoring times are $C_1,\dots ,C_n$ are i.i.d. Random Variable follows PDF $g_{\gamma }$ and CDF $G_{\gamma }$.

The PDF of the observation $Y_i$ is derived as $\begin{array}{rl}P(Y_i=t,{\delta }_i=0)& =P(C_i=t,T_i>C_i)=g_{\gamma }(t)S_{\theta }(t)\\ P(Y_i=t,{\delta }_i=1)& =P(T_i=t,T_i\le C_i)=f_{\theta }(t)(1-G_{\gamma }(t))\end{array}$ where $\theta$ is the parameter of interest, and $\gamma$ is the: nuisance parameter

Likelihood of Random Censoring Data

The Likelihood Function of random censored data is defined as $L(\theta )=\prod _{i=1}^n[f_{\theta }(t)(1-G_{\gamma }(t)){]}^{{\delta }_i}[S_{\theta }(t)g_{\gamma }(t){]}^{1-{\delta }_i}=c\cdot \prod _{i=1}^n{f}_{\theta }(t{)}^{{\delta }_i}{S}_{\theta }(y_i{)}^{1-{\delta }_i}$ where $c$ is a constant.