Kernel Estimation for Survival Analysis

Definition

Kaplan-Meier estimator is a step function. So it is difficult to calculate its quantile function and Density Function. The Kernel Density Estimation is used to make it smooth function.

Let $T_1,T_2,\dots ,T_n$ be i.i.d. survival time with a distribution $F$, and $C_1,C_2,\dots ,C_n$ be i.i.d. censoring time with a distribution $G$. 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.

Without Censoring

For the complete data, the Kernel Density Estimation is defined as $\hat{f}(t)=\frac{1}{n}\sum _{i=1}^n{K}_{\lambda }(t-Y_i)=\frac{1}{n\lambda }\sum _{i=1}^{n}K\left(\frac{t-Y_i}{\lambda }\right)$ where $K\text{s.t.}{\int }_{-\infty }^{\infty }K(t)dt=1,{\int }_{-\infty }^{\infty }{K}^2(t)dt<\infty$ is the kernel, $K_{\lambda }(x):=\frac{1}{h}K(\frac{x}{h})$ is the scaled kernel, and $\lambda$ is a smoothing parameter.

The kernel estimator for the Distribution Function is defined as $\hat{F}(t)={\int }_{-\infty }^t\hat{f}(u)du=\frac{1}{n}\sum _{i=1}^{n}W\left(\frac{t-Y_i}{\lambda }\right)$ where $W(t)={\int }_{-\infty }^{t}K(u)du$

With Censoring

For the censored data, the weights for each observation is defined as a jump size in Kaplan-Meier Estimator. $\hat{f}(t)=\sum _{i=1}^n{S}_i{K}_{\lambda }(t-Y_i)$ $\hat{F}(t)=\sum _{i=1}^n{S}_{i}W\left(\frac{t-Y_i}{h}\right)$ where $S_i$ is the jump size at $Y_i$ in Kaplan-Meier Estimator.

Thus, the Kernel Density Estimation for the Survival Function is $1-\sum _{i=1}^n{S}_{i}W\left(\frac{t-Y_i}{h}\right)$