Let $X_1,...,X_N$ denote $N$ independent random variables that follow a discrete uniform distribution $[0,T]$ with $T \in \mathbb{N}$ being the parameter we wish to estimate.
If we received a realization for each of these variables $x_1,...,x_N$, an unbiased estimator for $T$ is $\max\{x_1,...x_N\}\frac{N+1}{N}$.
But instead of getting a realization as a single shot, we receive a (partial) realization in multiple frames or steps.
In step 0, we receive all the realizations whose value is zero. In step 1, we delete one of the remaining variables and receive all the realizations whose value is 1 (but not the removed variable). In Step 2, we delete one of the remaining variables, and receive all the realizations whose value is equal to 2. Etc, etc, etc ...
Given this mechanism, we have that for some variables we DO know the exact realization. But for other variables, we only know a lower bound on the realization.
What would be an unbiased estimator for $T$?