A $$1 - \alpha$$ confidence interval for a parameter $$\theta$$ is an interval $$C_n = (a,b)$$ where $$a=a(X_1,\dotsc,X_n)$$ and $$b=b(X_1,\dotsc,X_n)$$ are functions of the data such that $$P(\theta \in C_n) = 1 - \alpha$$.

The action space is $$\mathcal{A}=\Theta$$.

## Loss function

$$L(\theta,a) = \left\{ \begin{array}{ll} 0 & \mbox{if } |\theta - a| \leq \delta \\ 1 & \mbox{if } |\theta - a| > \delta \\ \end{array} \right.$$

## Interpretation

A confidence interval is not a probability statement about $$\theta$$, as $$\theta$$ is a fixed quantity, not a random variable.

A $$1 - \alpha$$ confidence interval means that $$1 - \alpha$$ times when you construct confidence interval it will contain $$\theta$$.