Every competitor in a marathon has a unique number on their shirt, from 1 to N. N is unknown. The observation is $n_1, \ldots ,n_K$, which are randomly sampled from the $N$ competitors with equal probability. What is the MLE for $N$?
My intuition is that the MLE is simply the maximum observed in that set but how do I prove this? At first I thought this was a multinomial but that doesn't make sense since there is a single observation or there would be K observations without replacement. Is this a categorical distribution? How do I derive the MLE for that?
The probability that you get a particular subset of size $K$ from a population of size $N$ is $$ \frac 1 {\binom N K} = \frac{(N-K)!K!}{N!}. $$ This gets bigger as $N$ gets smaller. And $N$ can keep getting smaller until it reaches the maximum observed value; $N$ cannot be smaller than that.