I am reading Jun Shao's "Mathematical Statistics" second edition https://link.springer.com/book/10.1007/b97553. I need help understanding the statement and the proof of proposition 2.2 on page 117:
In the statement, $\mathbb{A}$ is the space of actions for our statistical decision and $A$ is a measurable subset of $\mathbb{A}$. The detailed setting can be found on the first paragraph of page 113 section 2.3.1. The definition of (randomized) decision rule can be found on the two paragraphs right above equation (2.22).
Let us write $(\Omega, M, \mu)$ for the probability space where $X$ is defined. I have the following questions:
- How exactly is $\delta_1$ defined? What is the right way of reading (2.23)? I find myself always having trouble with statements like "random variable given $T=t$". By the definition of decision rule, $\delta_0(\cdot, A)$ is a Borel function when $A \subseteq \mathbb{A}$ is a given measurable set. Since $X$ is a random variable defined $(\Omega, M, \mu)$ , we can view $Y=\delta_0(X, A)$ also as a random variable on $(\Omega, M, \mu)$. Does $E[\delta_0(X, A) \mid T=t]$ mean the conditional expectation of random variable $Y$ given the $\sigma$-algebra generated by the set $\{ T(X)=t \}$?
- If what I claim in 1 is correct, then how do I understand the first sentence in the proof? By definition of decision rule, we have to show that $\delta_1(\cdot,A)$ is a Borel function for every given $A$ and $\delta_1(t, \cdot)$ is a probability measure on $\mathbb{A}$. I can prove the latter but I am not sure about the first. It is not clear to me why this is related to sufficiency of $T$ either.
- If what I claim in 1 is correct, then how do I show the second equality? Let $C \in M$ be a $T(X)$ measurable set. It seems that I need to show:
\begin{equation*} \int_C \int_{\mathbb{A}} L(P,a) d\delta_1(x,a)d\mu(x) = \int_C \int_{\mathbb{A}} L(P,a) d\delta_0(x,a) d\mu(x) \end{equation*}
It is not clear to me why this holds.
- What is the importance of suddenly assuming $\mathbb{A}\subseteq \mathbb{R}^k$? Where is it used during the proof?