When estimating a vector of parameters, $\vec{\theta},$ based on observations of some random variables whose distribution depends on those parameters in some way, a confidence interval (for scalar $\theta$) or confidence region (for vector $\vec{\theta}$), is some set $\mathcal{C} = \mathcal{C}\left(X\right)$ such that $\mathcal{P}\left(\vec{\theta} \in \mathcal{C}\right) = 1 - \alpha$. To note:
- The confidence interval is a function of the data, $X$, so is itself random.
- The statement regarding the probability that $\vec{\theta}\in\mathcal{C}$ should be regarded with respect to the randomness in $X$ which controls $\mathcal{C}$. Since confidence intervals are a frequentist notion, one should not think of the probability as applying to the unobserved parameter $\vec{\theta}$, which, to a frequentist, is not random.
- A typical value for $\alpha$ is 0.05
- Often one can only compute approximate confidence intervals, which may have the nominal coverage asymptotically in the sample size.
