# Methods for estimating the survival function

I want to estimate the survival function $S(t_0)=1-F(t_0)$, at a specific point $t_0$ where $F$ is a continuous cumulative distribution function, based on an uncensored sample $x_1,..,x_n$ and a nonparametric estimator. So far, I am considering two estimators $\hat S(t_0)=1-\hat F(t_0)$, where $\hat F$ is

(1) Kernel density estimator

(2) The empirical CDF.

What other methods could I consider using?

## 3 Answers

Kaplan-Meier is the standard non-parametric survival estimator, which I'm guessing is what you meant by empirical CDF. You can get confidence intervals by various means with it so unless you want to go down a parametric route or start looking for more interesting things about the data I would say you're done!

• Thanks for the connection with the Kaplan-Meier estimator. – IwillSurvive Aug 18 '12 at 11:56
• Since the OP indicates there is no censoring, the K-M estimator corresponds exactly to subtracting the empirical cdf from 1. – cardinal Aug 18 '12 at 13:54
• I agree with you cardinal. – IwillSurvive Aug 18 '12 at 14:07
• Thanks, I am going to consider that I am not missing an alternative obvious estimator. – IwillSurvive Aug 18 '12 at 18:47

Kaplan-Meier and the smoothed version of the cdf are the only two nonparametric methods that come to mind. Many time in practice parametric models such as the negative exponential or the Weibull can be reasonable to assume. In such cases the parametric fit would be preferable.

Kaplan-Meier estimator is the standard, but you may also use Nelson-Aalen estimator. They are both consistent. If you need a smoothed estimator, you may first estimate the hazard function, then convert to a survival function.