Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.