Koul-Susarla-Van Ryzin Estimator

Definition

Let $T_1,T_2,\dots ,T_n$ be i.i.d. survival time, and $C_1,C_2,\dots ,C_n$ be i.i.d. censoring time. We can observe $(Y_i,{\delta }_i),i=1,2,\dots ,n$, where $Y_i=\min (T_i,C_i)$ and ${\delta }_i=I(T_i\le C_i)$ is censoring indicator

Suppose a Simple Linear Regression model $E(T_i)=\alpha +\beta x_i$

Koul-Susarla-Van Ryzin proposed an Unbiased Estimator for $\alpha +\beta x_i$ $Y_i^{\ast }=\frac{{\delta }_i{Y}_i}{1-G(Y_i)}$ Since $G(Y_i)$ is unknown, it should be estimated. Authors suggested to use Kaplan-Meier Estimator of $G$ with data $(Y_i,1-{\delta }_i),i=1,2,\dots ,n$ as an estimator.