Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$.

Multiple "samples" or observations for each of these variables are given (not necessarily with the same size).

For each of variable $X_i$ I define the estimator $\hat{X_i}=\sum_{k}\frac{r_k}{N_i}$ which is an unbiased estimator for the mean value. ($r_k$ is the $k^{th}$ sample and $N_i$ is the number of samples from variable $i$.)

Am I correct up to this point?

Defined $Z=\max\{X_1,X_2,...,X_n\}$. (The max of mean values)

What would make an unbiased estimator for $Z$?

How about $\hat{Z}=\max\{\hat{X}_1,\hat{X}_2,...,\hat{X}_n\}$?

I tried to calculate the bias of $\hat{Z}$ and got 0. Am I wrong?

Thanks.