Miller 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 $Y_i=\alpha +\beta X_i+{ϵ}_i,i=1,2,\dots ,n$

With no censoring present, the least squares estimators of the parameters $\alpha ,\beta$ are obtained by minimizing $\frac{1}{n}\sum _{i=1}^n{ϵ}_i^2=\frac{1}{n}\sum _{i=1}^n(T_i-\alpha -\beta x_i{)}^2={\int }_{-\infty }^{\infty }{z}^{2}d{F}_n(z)$ where $F_n(z):=\frac{1}{n}\sum _{i=1}^{n}I({ϵ}_i\le z)$ is the Empirical Distribution Function of $z_1,z_2,\dots ,z_n$ where $z_i=y_i-\alpha -\beta x_i$.

With censoring present, Miller proposed to minimize ${\int }_{-\infty }^{\infty }{z}^{2}d{\hat{F}}_n(z)=\sum _{i=1}^n{\hat{w}}_i(\beta )(Y_i-\alpha -\beta x_i{)}^2$ where $\hat{F}$ is the Kaplan-Meier Estimator based on $(z_i,{\delta }_i)$ and the weights ${\hat{w}}_i(\beta )$ is its jump size.

If the last observation is censored, then $\sum _{i=1}^n{\hat{w}}_i(\beta )<1$. Hence, change the last observation to be uncensored, so that $\sum _{i=1}^n{\hat{w}}_i(\beta )=1$.