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

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

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

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

P(gL(X‾)≤θ,gU(X‾)≥θ)=P(θ∈[gL(X‾),gU(X‾)])\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 [gL(X‾),gU(X‾)]\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 θ:\theta:

inf⁡θ∈ΘP(θ∈[gL(X‾),gU(X‾)])\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 θ\theta and therefore the coverage probability and confidence coefficient are equal.

If the interval estimator [gL(X‾),gU(X‾)]\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right] is designed with confidence coefficient 1−α1-\alpha for some α∈(0,1):\alpha\in\left(0,1\right):

P(θ∈[gL(X‾),gU(X‾)])≥1−α  ∀ θ∈Θ\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 [gL(X‾),gU(X‾)]\left[g_L\left(\underline{X}\right),g_U\left(\underline{X}\right)\right] is a 100(1−α)%100\left(1-\alpha\right)\% confidence interval.