Buckley-James 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

If we can observe the true survival time $T_i$, we can make a model $E(T_i)=\alpha +\beta x_i$ However, we can’t observe $T_i$, but only censored $Y_i$, and $E(Y_i)\ne \alpha +\beta x_i$ Buckley and James proposed an Unbiased Estimator for $\alpha +\beta x_i$ $Y_i^{\ast }=Y_i{\delta }_i+E(T_i|T_i>Y_i)(1-{\delta }_i)$ Since we also can not observe $y_i^{\ast }$, we estimate it again. ${\hat{y}}_i^{\ast }=y_i{\delta }_i+\left[\hat{\beta }{x}_i+\frac{\sum _{k:{\hat{z}}_k>{\hat{z}}_i}{\hat{w}}_k(\hat{\beta }){\hat{z}}_k}{1-\hat{F}({\hat{z}}_i)}\right](1-{\delta }_i)$ where ${\hat{z}}_i=y_i-\hat{\beta }{x}_i$, $\hat{F}$ is the Kaplan-Meier Estimator based on $({\hat{z}}_i,{\delta }_i)$ and the weights ${\hat{w}}_i(\hat{\beta })$ is its jump size.

The variance of the estimator is estimated by $\widehat{\mathrm{Var}}(\hat{\beta })=\frac{{\hat{\sigma }}_u^2}{\sum _{i:{\delta }_i=1}(x_i-{\bar{x}}_u{)}^2}$ where ${\hat{\sigma }}_u^2=\frac{1}{n_u-2}\sum _{i:{\delta }_i=1}[y_i-{\bar{y}}_u-\hat{\beta }(x_i-{\bar{x}}_u){]}^2$, $n_u=\sum _{i=1}^n{\delta }_i$, ${\bar{x}}_u=\sum _{i:{\delta }_i=1}{x}_i$, and ${\bar{y}}_u=\sum _{i:{\delta }_i=1}{y}_i$