Nonparametric inference
Survival data is subject to censoring and we need to introduce a new quantity to capture this information. Suppose that we have a n IID random sample of lifetime random variables in 
, and we know that the event of interest might have occurred or that it will occur sometime in the future. The additional information is captured by the Kronecker indicator variable, 
:
Thus, we have n pairs of random observations in the Ts and 
s, 
. To obtain the estimates of the cumulative hazard function and the survival function, we will need an additional notation. Let 
denote the unique times of Ts at which the event of interest is observed. Next, we denote 
to represent the number of observations that are at risk just before times 
and 
the number of events that occur at that time. Using these quantities, we now propose to estimate the cumulative hazard function using the following:
The estimator 
is the famous Nelson-Aalen estimator. The Nelson-Aalen estimator enjoys statistical properties...
