Nelson-Aalen Estimator

Definition

Suppose that $T_1,T_2,\dots ,T_n$ are i.i.d. random variables represent survival times with CDF $F$, and $C_1,C_2,\dots ,C_n$ are the censoring times. And let ${\delta }_i=\{\begin{array}{ll}1& T_i\le C_i\text{: uncensored}\\ 0& T_i>C_i\text{: censored}\end{array}$ be the censoring indicator. In a Random Censoring setting, we only observe $(Y_i,{\delta }_i),i=1,\dots ,n$ where $Y_i=\min (T_i,C_i)$.

The Nelson-Aalen estimator estimates cumulative hazard function as $\hat{\Lambda }(t)=\sum _{i:y_i\le t}\frac{{\delta }_i}{(n-i+1)}$ It is derived from Kaplan-Meier Estimator