Solution: Special case gives Nelson-Aalen

We have counting processes \(N_i(t)\) for \(i = 1,2,\ldots,n\), with intensity processes of the form \(\lambda_i(t) = Y_i(t)\beta_0(t)\), where the \(Y_i(t)\) are at risk indicators. This is a special case of the additive regression model \((4.55)\), with

\[\mathbf{N}(t) = (N_1(t),\ldots,N_n(t))^T,\]

\[\mathbf{B}(t) = B_0(t) = \int_0^t\beta_0(u)du,\]

\[\mathbf{X}(t) = (Y_1(t),\ldots,Y_n(t))^T.\]

Note that the design matrix \(\mathbf{X}(t)\) only contains the linear term (with respect to what would be the covariates), since there are no covariates in the model. \(\mathbf{X}(t)\) has full rank unless all the \(Y_i(t)\) are 0, which only happens for \(t\) larger than the last event time. Thus for all event times \(T_j\) we will have \(J(T_j) = 1\).

Further we have

\begin{align*} \mathbf{X^-}(t) &= (\mathbf{X}(t)^T\mathbf{X}(t))^{-1}\mathbf{X}(t)^T \\ &= ((Y_1(t),\ldots,Y_n(t))(Y_1(t),\ldots,Y_n(t))^T)^{-1}(Y_1(t),\ldots,Y_n(t)) \\ &= \frac{1}{\sum_{i=1}^n Y_i(t)^2}(Y_1(t),\ldots,Y_n(t)) \\ &= \frac{1}{\sum_{i=1}^n Y_i(t)}(Y_1(t),\ldots,Y_n(t)) \\ &= \frac{1}{Y(t)}(Y_1(t),\ldots,Y_n(t)),\end{align*}

where \(Y(t)\) is the risk indicator of the aggregated process \(N(t) = \sum_{i=1}^nN_i(t)\).

We also have that \(\Delta \mathbf{N}(T_j) \) is a vector of zeros except for a one for the component corresponding to the individual who experiences the event at time \(T_j\), so that \((5.49)\) gives us

\begin{align*} \widehat{B_0}(t) &= \sum_{T_j \leq t} J(T_j)\mathbf{X^-}(T_j)\Delta \mathbf{N}(T_j) \\ &= \sum_{T_j \leq t} 1 \cdot \frac{1}{Y(T_j)}(Y_1(T_j),\ldots,Y_n(T_j)) \cdot (0,\ldots,1,\ldots,0)^T \\ &= \sum_{T_j \leq t} \frac{Y_{i_j}(T_j)}{Y(T_j)} \\ &= \sum_{T_j \leq t} \frac{1}{Y(T_j)},\end{align*}

which is exactly the Nelson-Aalen estimator from \((3.4)\).