In our intro statistic lecture the following we said that the following components made up an estimation problem
an at most countable space $\mathcal{X}$ of all possible samples we can observe
a family $(P_\theta)_{\theta \in \Theta} $ of distributions on $\mathcal{X}$
a function $f:\Theta \rightarrow \mathbb{R}$ that we would like to estimate
Then we discussed an example were the aim was to find out how many cabs there are in this city: We assumed that
-> all cabs within the city are numbered with numbers from the set $\{1,\ldots,M\}$
-> we observe cabs with numbers $t_1<\ldots<t_m$
The problem I have with this example is that we said that $\mathcal{X}$ is the set of all subsets of size $m$ of $\{1,\ldots,M\}$. What bothers me about that, is that $M$ is a parameter it $\Theta=\mathbb{N}$, which we wish to estimate, so our space $\mathcal{X}$ actually depends on it: Thus we actually have a family $(\mathcal{X}_M)_{M\in \Theta}$ instead of a fixed $\mathcal{X}$.
Is that allowed ? I thought in the general definition above that $\mathcal{X}$ shouldn't depend on $\theta$; otherwise it should have been written $(\mathcal{X}_\theta)_{\theta\in \Theta}$.
