Ask the notation on confidence sets

Question

Theorem 9.2.2: For each $\theta_0 \in \Theta$, let $A(\theta_0)$ be the acceptance region of a level $\alpha$ test of $H_0:\theta=\theta_0$. For each $\textbf{x} \in X$, define a set $C(\textbf{x})$ in the parameter space by

$$C(\textbf{x})= \{\theta_0: \textbf{x} \in A(\theta_0) \}$$

Then the random set $C(\textbf{x})$ is a $1-\alpha$ confidence set.

Okay, my question is to focus on the notation of $C(\textbf{x})$. I am wondering since $C(\textbf{x})$ is studying the unknown parameter, why don't we write down $C(\theta)$?

Answer 1

Probably because the confidence interval/set depends on the data. If you get an observation of say $\textbf x$ you would get one confidence set for $\theta$ and a different observation of the random variable $X$ gives you a different confidence set. $(1-\alpha)\times 100\%$ of those will contain the parameter $\theta$.

For example, if you were estimating heights and weights of people in a city, one such confidence set from a set of $n_1$ observations would be different from a confidence set from another set of $n_2$ observations.

Note: Confidence set is confidence interval for a multidimensional parameter.