Confidence band ($\mathbb{P}(L(t)\le f(t)\le U(t)\text{ for every }t)\ge 1-\alpha$)

A confidence band provides simultaneous bounds on an unknown function over its entire index set. Its coverage event is the intersection over all indices, unlike separate pointwise confidence intervals. For a continuous cumulative distribution function, the Kolmogorov-Smirnov theorem gives an asymptotic band $F_n\pm c_{\alpha }/\sqrt{n}$, clipped to $[0,1]$, where $c_{\alpha }$ is a $(1-\alpha )$-quantile of the absolute Brownian bridge maximum. The simultaneous event equals $\sqrt{n}\Vert F_n-F{\Vert }_{\infty }\le c_{\alpha }$, so convergence in distribution at this continuity point proves coverage tending to $1-\alpha$.

For independent observations from a continuous distribution function, the probability integral transform makes the Kolmogorov-Smirnov statistic distribution-free. Choose $c_{n,\alpha }$ from its uniform-sample distribution to have probability $1-\alpha$ of not exceeding that critical value. Clipping the empirical distribution plus/minus this value to the unit interval gives simultaneous coverage at every argument. This is a band for the entire distribution, rather than separate pointwise intervals.

// Target: statistical-modelling.bigb