# Rewriting integration by summation

#### user1234

##### New Member
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?