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As seeing the comment on the 2nd answer of this question, mentioning delta function, I tried to write the proof of that part.

Proof

As $T(\mathbf{x})=t\mathbf{1}_{\mathbf{X}}(\mathbf{x}) \Rightarrow \mathbf{X} = \mathbf{x}$, $\mathbf{X}=\mathbf{x}$ is necessary to $\mathbf{T}(\mathbf{x})=t$.

Therefore, $\{\{\omega \mid\mathbf{X}=\mathbf{x}\} \cap \{\omega \mid\mathbf{T}(\mathbf{x})=t\}\}=\{\omega \mid\mathbf{X}=\mathbf{x}\}$.

$\mathbf{1}_{\mathbf{X}}(\mathbf{x})$ is an indicator function, which is defined

$$ {\displaystyle \mathbf {1} _{X}(x):={\begin{cases}1~&{\text{ if }}~x\in X~,\\0~&{\text{ if }}~x\notin X~.\end{cases}}} $$

Question

Is this correct? Note that I am not used to writing a rigorous proof in English so some notations might be awkward.

CatsEyeblow
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  • What are you trying to prove? $P_{\theta}(X=x,T=t)=P_{\theta}(X=x)$ is not true in general unless ${X=x}\subset {T=t}$, in which case this is just a general fact: $$P(A\cap B) = P(A),\quad A\subset B$$, as $A\cap B = A$ – Andrew Jul 15 '23 at 19:09
  • Thanks, @AndrewZhang. I really appreciate your comment. Let me ask you a question. Is it always true that $P_{\theta}(X=x,T=t)=P_{\theta}(X=x)$ if there is an assumption of $T(x) = t$, because $T$ is derives from $X$ and we can say that ${X=x}\subset {T=t}$? That is the only thing that my textbook says about the assumption of this theorem. – CatsEyeblow Jul 15 '23 at 21:26
  • It may make things clearer if you write $T=f(X)$, for some $f$. Then, provided $x\in {u:f(u)=t}$, one has ${ X=x}\subset {T=t}$. – Andrew Jul 15 '23 at 21:52
  • Yes, that is clearer and more obvious. Thank you! – CatsEyeblow Jul 15 '23 at 22:10

1 Answers1

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Please see the comment of the question. Thanks to Andrew, I understand what I was wrong and where to start.

Summary

Given $T(\mathbf{x})=t$, $T$ can be written as $T=f(X)$, for some $f$.

Then for $x\in \{u:f(u)=t\}$, one has $\{ X=x\}\subset \{T=t\}$.

CatsEyeblow
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  • Your answer could be improved with additional supporting information. Please [edit] to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. – Community Jul 15 '23 at 22:58