Reduced Sample Estimator

Definition

Notations

Assume intervals have equal length and let the notations:

• $I_i$: the $i$-th interval $({\tau }_{i-1},{\tau }_i]$ where ${\tau }_0=0$
• $n_i$: the number of individuals alive at the beginning of $I_i$
• $d_i$: the number of deaths during $I_i$
• $l_i$: the number of individuals censored during $I_i$
• $P_i$: $P(T>{\tau }_i|T>{\tau }_{i-1})=P(\text{surviving through}{I}_i|\text{alive at beginning of}{I}_i)$ where $T_i$ is the survival time.
• $q_i=1-P_i$

Estimation of Reduced Sample Estimator

Reduced sample method estimates the Survival Function as $\hat{S}({\tau }_k)=1-\frac{d_k^{\ast }}{n_k^{\ast }}$ where $d_k^{\ast }=\sum _{i=1}^k{d}_i$ is the number of deaths in the interval $(0,{\tau }_k)$, and $n_k^{\ast }=n_1-\sum _{i=1}^k{l}_i$ is the number of uncensored data in $(0,{\tau }_k)$

The drawback of the reduced sampling method is that it ignore the information contained in censored observations, therefore it usually a biased (under) estimator of the Survival Function $S(t)$.