Kendal Day

2022-04-08

Trouble Understanding the Formal Definition of a Confidence Interval

A$1-\alpha$ confidence interval for a paramater $\theta$ is an interval ${C}_{n}=(a,b)$ where $a=a({X}_{1},\cdots ,{X}_{n})$ and $b=b({X}_{1},\dots ,{X}_{n})$ are functions of the data such that

$\mathbb{P}}_{\theta}(\theta \in {C}_{n})>1-\alpha \text{, for all}\text{}\theta \in \mathrm{\Theta$

If$\theta$ is a vector then we use a confidence set (such as a sphere or an ellipse) instead of an interval.

Question:

While I understand conceptually what a confidence interval is (i.e., a 95% CI means that 95% of experiments will trap the paramater in the interval), I don't understand how this formality is capturing this concept.

In particular, I don't understand what is meant by the notion of${\mathbb{P}}_{\theta}(\theta \in {C}_{n})$ . What is the sample space which P is drawing from? What is the set $\theta \in {C}_{n}$ ? It seems here $\theta$ is being treated both as a fixed value (from the notation Pθ) and as a random variable (by the notation $\theta \in {C}_{n}$ ).

A

If

Question:

While I understand conceptually what a confidence interval is (i.e., a 95% CI means that 95% of experiments will trap the paramater in the interval), I don't understand how this formality is capturing this concept.

In particular, I don't understand what is meant by the notion of

vadomosigmx6

Beginner2022-04-09Added 12 answers

Viewed this way, the event (1) is more a statement about the random endpoints a and b, than about the parameter

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