Estimators for Survival Function

Definition

Mean of Survival Time

Without Censoring

$\hat{\theta }={\int }_0^{\infty }xd{F}_n(x)=\bar{x}$ where $F_n$ is the empirical CDF

With Censoring

$\hat{\theta }={\int }_0^{\infty }xd\hat{F}(x)={\int }_0^{\infty }\hat{S}(t)dt=\sum _{i=1}^n{S}_i{Y}_i$ where $\hat{F}$ is calculated by the Kaplan-Meier Estimator and $S_i$ is the jump size at $Y_i$.

The asymptotic variance of the parameter is obtained by $\mathrm{AVar}(\hat{\theta })=\frac{1}{n}{\int }_0^{\infty }\frac{1}{(1-H(S){)}^2}{\left({\int }_0^{\infty }S(t)dt\right)}^{2}d{F}_u(S)$ And the asymptotic variance is estimated by $\widehat{\mathrm{AVar}}(\hat{\theta })=\sum _{i=1}^n{\left({\int }_{y_i}^{\infty }S(t)dt\right)}^2\frac{{\delta }_i}{(n-i)(n-i+1)}$ where $(1-\widehat{H(S)}{)}^2=\frac{n-i+1}{n}$ and $d{\hat{F}}_u(y_i)=\frac{{\delta }_i}{n}$

Median of Survival Time

A reasonable estimator for $\theta$ is $\hat{\theta }={\hat{S}}^{-1}(0.5)$ where $\hat{S}$ is the Kaplan-Meier Estimator

If ${\hat{S}}^{-1}$ does not have a unique solution, then $\hat{\theta }$ is defined as the midpoint of the interval constituting of the solutions.

However, the estimator $\hat{S}(t)$ over-estimate the true parameter, so linear smooth of the estimator $\hat{\hat{S}}(t)$ is used to estimate the $\theta$ $\hat{\hat{\theta }}={\hat{\hat{S}}}^{-1}(0.5)$

The asymptotic variance of the estimated parameter is obtained by $\mathrm{AVar}(\hat{\theta })=\frac{\mathrm{AVar}(\hat{S}(\theta ))}{f^2(\theta )}$ where $\mathrm{AVar}(\hat{S}(\theta ))$ is estimated by Greenwood’s formula, and $f$ may estimated by Kernel Estimation.