I'm just starting to learn about Survival Analysis and I understand the theoretical proofs but I'm still unclear about how to do some of the estimations practically.
The primary complication in the analysis of survival data is censoring, i.e., the fact that you do not observe the event of interest for some subjects in your study. Many statistical procedures do not work with censored data, such as the empirical cumulative distribution function.
To obtain valid estimates you will need to use statistical techniques that do account for it. If you have right censored data, you can obtain a non-parametric estimate of the cumulative hazard function using the Nelson-Aalen estimator. Another option is to first obtain an estimate of the survival function using the Kaplan-Meier estimator and then calculate the minus logarithm of it to obtain the cumulative hazard function.