Suppose that n individuals have lifetimes represented by random variables T_1, T_2, \ldots, T_n. Instead of the observed values for each lifetime, we have a time t_i' which we know is either the lifetime or censoring time.

Let us define a variable \delta_i=I(T_i=t_i') that equals 1 if T_i=t_i' and 0 if T_i>t_i'. This \delta_i is called the censoring or status indicator for t_i', since it tells us if t_i' is an observed lifetime (\delta_i=1) or censoring time (\delta_i=0). The observed data then consist of (t_i',\delta_i), i=1,2,\ldots n.

Suppose that there are k (k\le n) distinct times t_1<t_2<\ldots t_k at which death occurs. The possibility of there being more than one death at t_j is allowed, and we let d_j=\sum I(t_i'=t_j,\delta_i=1) represents the number of deaths at t_j. In addition to the lifetimes t_1,\ldots, t_k, there are also censoring times for individuals whose lifetimes are not observed.

Also let n_j=\sum I(t_i'\ge t_j) is the number of individuals at risk at t_j.

The Nelson-Aalen estimator is given by:

\hat H(t)=\int_{0}^{t}\frac{dN(u)}{Y(u)}=\int_{0}^{t}\frac{d\sum_{i=1}^{n}N_i(u)}{\sum_{i=1}^{n}Y_i(u)}
\Rightarrow\hat H(t)=\sum_{j:t_j\le t}\frac{d_j}{n_j}\ldots (1)

With a hypothetical example, let me show how equation (1) works:

\begin{array}{l|cccccccccc}
t_i' & 6 & 6 & 6 & 7 & 9 & 10 & 10 & 11 & 13 & 16 \\
\hline
\delta_i & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1\\
\end{array}

From the above data, we can calculate \hat H(t) by following:

\begin{array}{c|c|c|c}
t_j & n_j & d_j & \hat H(t_j) \\
\hline
6 & 10 & 2 & 0.2 \\
7 & 7 & 1 & 0.343 \\
9 & 6 & 0 & 0.343\\
10 & 5 & 1 & 0.543 \\
11 & 3 & 0 & 0.543\\
13 & 2 & 1 & 1.043\\
16 & 1 & 1 & 2.043
\end{array}


Incorporating a weight function w_i(u),i=1,2,\ldots n, the estimator for the cumulative hazard function is

\hat H_w(t)=\int_{0}^{t}\frac{\sum_{i=1}^{n}w_i(u)dN_i(u)}{\sum_{i=1}^{n}w_i(u)Y_i(u)}. \ldots (2)

Now I want to rewrite \hat H_w(t) of equation (2) in the representation of summation as like equation (1).

So I wrote it as

\hat H_w(t)=\int_{0}^{t}\frac{\sum_{i=1}^{n}w_i(u)dN_i(u)}{\sum_{i=1}^{n}w_i(u)Y_i(u)}
\Rightarrow\hat H_w(t)=\sum_{j:t_j\le t}\frac{w_jd_j}{w_jn_j}\ldots (3)

But in equation (3), w_j of numerator and denominator cancels out and it reduces to equation (1). That is, I couldn't correctly rewrite equation (2).

How can I express \hat H_w(t) of equation (2) in the representation of summation?