Estimate population size based on first repeat https://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation asks the following question.

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?

The provided answer gives a maximum likelihood estimator as well as an unbiased estimator of $k(k+1)/2$  
If you know the number of buses can't be larger than some given value $N \geq k+1$, how does that change the maximum likelihood estimator and the unbiased estimator ?
 A: The unbiased estimator doesn't change. The requirement that the expected value of the estimate is $n$ for all $N$ possible values of $n$ yields $N$ linear contraints on the $N$ estimates of $n$ for the $N$ different possible values of $k$. This $N\times N$ system of linear equations is triangular with non-zero diagonal and thus has a unique solution; since we know that $k(k+1)/2$ is a solution, this is the only unbiased estimator. It yields estimates of $n$ greater than $N$ for most values of $k$, but there is no unbiased estimator without this undesirable property. This is also intuitively clear, since for $n=N$ the estimates below $N$ must be compensated by estimates above $N$ to obtain the expected value $N$.
The maximum likelihood estimator changes in that if the original maximum likelihood estimate would have been greater than $N$, this should be replaced by $N$, since the likelihood is unimodal and thus takes its maximum on $\{1,\dotsc,N\}$ at $N$ if the maximum likelihood estimate is greater than $N$.
