## Rewriting integration by summation

Suppose that individuals have lifetimes represented by random variables Instead of the observed values for each lifetime, we have a time which we know is either the lifetime or censoring time.

Let us define a variable that equals if and if . This is called the censoring or status indicator for , since it tells us if is an observed lifetime or censoring time The observed data then consist of

Suppose that there are distinct times at which death occurs. The possibility of there being more than one death at is allowed, and we let represents the number of deaths at In addition to the lifetimes there are also censoring times for individuals whose lifetimes are not observed.

Also let is the number of individuals at risk at

The Nelson-Aalen estimator is given by:

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

From the above data, we can calculate by following:

Incorporating a weight function , the estimator for the cumulative hazard function is

Now I want to rewrite of equation in the representation of summation as like equation .

So I wrote it as

But in equation , of numerator and denominator cancels out and it reduces to equation . That is, I couldn't correctly rewrite equation .

How can I express of equation in the representation of summation?