 Solution

Let $D={\sum }_{j=1}^n{1}_{\{T_j\le c_j\}}$ be the observed event count. For each subject, the fixed-censoring identity gives $\mathbb{E}H(X_j)=\mathbb{P}(T_j\le c_j)$. Linearity of expectation therefore yieldsIndependence 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.