Confidence Interval

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

$t\in A(\vartheta )⟺\vartheta \in CI(t).$

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:

$A(\vartheta )=\left\{t:p_{\vartheta }(t)\ge \alpha \right\}⟺Cl(t)=\left\{\vartheta :p_{\vartheta }(t)\ge \alpha \right\}$

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

• Two-Sided Gauß Test
• One-Sided Gauß Test
• Two-Sided t-Test
• One-Sided t-Test
• Wald-CI
• Agresti-Cull-CI
• Asymptotic CI for Variance

Estimate Quantiles of CDF by quantiles of ECDF

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

With $k=\lceil np-z_{1-\alpha /2}\sqrt{np(1-p)}\rfloor ,\ell =\lfloor np+z_{1-\alpha /2}\sqrt{np(1-p)}\rfloor$

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

This approximation is good for $np(1-p)\ge 10.$