# Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

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 $(\mathcal{X}_\theta)_{\theta\in \Theta}$.

-
The sample space should be the set of all subsets of size $m$ of $\mathbb{N}$, i.e., the union of the sample spaces for each fixed number of cabs. For each $\theta \in \Theta$, you have a subset of these samples that are consistent with that $\theta$ (the support of $P_{\theta}$). – mjqxxxx Jan 2 '13 at 15:38