This is a follow up to Estimate population size based on first repeat as follows.
The previous question is as follows.
I take the bus to work every day. Every bus has a serial number, but unlike in the German Tank Problem, I don't know if they are numbered uniformly 1...n.
Suppose the first k buses are all different, but on day k+1 I take one I've been on before. What is the best estimate for the total number of buses?
An unbiased estimator of $k(k+1)/2$ is given as well as a MLE estimate. If you want to get a more accurate estimate it seems to make sense to wait until the second or $m$th repeat, for some fixed $m$, and then stop. How would you calculate the variance of your new estimate?