= Solution
Let $D=\sum_{j=1}^n\mathbf1_{\{T_j\le c_j\}}$ be the observed event count. For each subject, the fixed-censoring identity gives $\mathbb EH(X_j)=\mathbb P(T_j\le c_j)$. <Linearity of expectation> therefore yields
$$
\boxed{\mathbb E\sum_{j=1}^nH(X_j)=\sum_{j=1}^n\mathbb P(T_j\le c_j)=\mathbb ED.}
$$
<Independence> between subjects is not needed for this <expectation> identity. The common $H$ presumes a common marginal <hazard>; with heterogeneous subject-specific <hazards>, replace it by $H_j$ for subject $j$. Random <censoring> admits the analogous conditional argument when its <independence> assumptions hold.
Back to article page
