Confidence Interval

The Confidence Interval is more or less the inverse of an Acceptance Domain

t \in A(\vartheta) \Longleftrightarrow \vartheta \in C I(t).

This means it contains the most probable parameters given some $t$ from the Acceptance Domain.

From a level-alpha test with p-Value one can easily derive the Confidence Interval like this:

A(\vartheta)=\left\{t: p_{\vartheta}(t) \geq \alpha\right\} \Longleftrightarrow C l(t)=\left\{\vartheta: p_{\vartheta}(t) \geq \alpha\right\}

General Definition

We have some sample $X_1,\dots, X_n$ from a distribution with parameters $\vartheta \in \Theta$ where $\vartheta$ can for example be the Expectation of a Normal Distribution.
Then we can use the sample to create the Random Variables $U$ and $O$ which as an interval $[U,O]$ make up the random interval of coverage propability

\mathbb{P}_{\vartheta}(\vartheta \in[U, O])

It is called the $1-\alpha$ confidence interval for $\vartheta$ if

\mathbb{P}_{\vartheta}(\vartheta \in[U, O]) \geq 1-\alpha

We want to choose confidence intervals as small as possible.

Using a concrete data sequence, one can say that in about $1-\alpha$ percent of the cases the parameter $\vartheta$ will lie in the interval.

Examples

See:

Estimate Quantiles of CDF by quantiles of ECDF

\mathbb{P}\left(x_p \in\left[X_{(k)}, X_{(\ell)}\right]\right) \geq B(n, p)([k, \ell))

With

k=\left\lceil n p-z_{1-\alpha / 2} \sqrt{n p(1-p)}\right\rfloor, \quad \ell=\left\lfloor n p+z_{1-\alpha / 2} \sqrt{n p(1-p)}\right\rfloor

We get an asymptotically exact $1-\alpha$ CI for the quantile $x_p$ .

This approximation is good for

np(1-p)\geq 10.