Sum over specific values

Question

I'm asked to simulate 50 survival times from an exponential distribution with rate 1.

n <- 50
Tstar <- rexp(n, rate = 1)

Then I have the following quantities:

Y(t) capturing the individuals at risk at time t, i.e.

Y <- function(t){sum(Tstar > t)}

and S(t) is the Kaplan-Meier estimator

S <- function(t)(1 - 1/n * sum(Tstar < t)

But how do I define the following function?

Here Tstar[i] indicates T_i.

Answer 1

If I understand my mathematical notations, consider Reduce + sapply to serve as iterator and summation across specific values:

set.seed(4621)

n <- 50
Tstar <- rexp(n, rate = 1)

Y <- function(t) sum(Tstar > t)
S <- function(t) (1 - 1/n * sum(Tstar < t))

sigma_sq <- function(t) {
  Tstar <- Tstar[Tstar < t]
  S(t)^2 * Reduce(`+`, sapply(Tstar, function(T_i) 1/(Y(t)*(T_i)^2)))
}