Let \(\underline{X}\) be a random sample of size \(n\) distributed according to the population \(\mathbb{P}_{X;\,\theta},\) which depends on the parameter \(\theta,\) and common state space \(S.\) An interval estimator for \(\theta\) is a random interval \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\) where the statistics \(g_L\left(\underline{X}\right)\leq g_U\left(\underline{X}\right)\,\,\forall\,\underline{X}\in S^n\) are intended to contain the true parameter \(\theta.\)

An interval estimate for the parameter \(\theta\) is obtained by using the realised values of the statistics \(g_L\left(\underline{x}\right)\) and \(g_U\left(\underline{x}\right)\) and is equivalent to stating \(g_L\left(\underline{x}\right)\leq\theta\leq g_U\left(\underline{x}\right).\)

A one-sided interval estimate is of the form \(\left(-\infty,g_U\left(\underline{x}\right)\right]\) or \(\left[g_L\left(\underline{x}\right),\infty\right)\) and is equivalent to stating \(\theta\leq g_U\left(\underline{x}\right)\) or \(\theta\geq g_L\left(\underline{x}\right)\) respectively.

Let \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\) be an interval estimator for the parameter \(\theta.\) The coverage probability is the probability that \(\theta\) lies within \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right].\) The coverage probability is equivalent to the joint probability:

\[\begin{align} \mathbb{P}\left(g_L\left(\underline{X}\right)\leq\theta,g_U\left(\underline{X}\right)\geq\theta\right)\\ =\mathbb{P}\left(\theta\in\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\right) \end{align}\]

The confidence coefficient of the interval estimator \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\) is the infimum of the coverage probability over all possible values of the true parameter \(\theta:\)

\[\inf_{\theta\in\Theta}\mathbb{P}\left(\theta\in\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\right)\]

The coverage probability is often constant as a function of \(\theta\) and therefore the coverage probability and confidence coefficient are equal.

If the interval estimator \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\) is designed with confidence coefficient \(1-\alpha\) for some \(\alpha\in\left(0,1\right):\)

\[\mathbb{P}\left(\theta\in\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\right)\geq 1-\alpha\,\,\forall\,\theta\in\Theta\]

then \(\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right]\) is a \(100\left(1-\alpha\right)\%\) confidence interval.